Continuity and Differentiability QRT – Class 12 Maths (Free Download)

Continuity and Differentiability QRT – Quick Revision Tables bring the important concepts, formulas and results of the chapter together in one place. Use these tables for a quick recap while revising, practising questions or preparing for your exams.

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Table 1: Continuity and Differentiability QRT – Continuity Basics

Concept	Formula / Result
Continuity at a Point	A function \displaystyle f is continuous at \displaystyle x=c if \displaystyle \lim_{x\to c}f(x)=f(c)
Three Conditions	For continuity at \displaystyle x=c: (i) \displaystyle f(c) must exist, (ii) \displaystyle \lim_{x\to c}f(x) must exist, and (iii) \displaystyle \lim_{x\to c}f(x)=f(c)
Left & Right Limits	Continuity at \displaystyle x=c requires \displaystyle \lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c)
Continuity on Domain	A real function is continuous if it is continuous at every point in its domain.
Interior Point	At an interior point \displaystyle x=c, both sides belong to the domain, so \displaystyle \lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c)
Left Endpoint	If \displaystyle f is defined on \displaystyle [a,b], continuity at \displaystyle a means \displaystyle \lim_{x\to a^+}f(x)=f(a)
Right Endpoint	If \displaystyle f is defined on \displaystyle [a,b], continuity at \displaystyle b means \displaystyle \lim_{x\to b^-}f(x)=f(b)
Singleton Domain	If the domain of \displaystyle f consists of only one point, then \displaystyle f is continuous at that point.
Discontinuity	If \displaystyle \lim_{x\to c}f(x)\ne f(c), then \displaystyle f is not continuous at \displaystyle x=c
Constant Function	The constant function \displaystyle f(x)=k is continuous at every real number.
Identity Function	The identity function \displaystyle f(x)=x is continuous at every real number.
Graphical Check	Graphically, a function is continuous at a point if its graph has no break, jump or hole at that point.

Maths Better Tip… Remember: to check continuity at a point, always compare the left-hand limit, right-hand limit and the function value. All three must be equal. Graphically, think of continuity as the graph having no break, jump or hole at the point.

Table 2: Continuity and Differentiability QRT – Algebra & Discontinuity

Concept	Formula / Result
Sum	If \displaystyle f and \displaystyle g are continuous at \displaystyle c, then \displaystyle f+g is continuous at \displaystyle c.
Difference	If \displaystyle f and \displaystyle g are continuous at \displaystyle c, then \displaystyle f-g is continuous at \displaystyle c.
Product	If \displaystyle f and \displaystyle g are continuous at \displaystyle c, then \displaystyle fg is continuous at \displaystyle c.
Quotient	If \displaystyle f and \displaystyle g are continuous at \displaystyle c and \displaystyle g(c)\ne0, then \displaystyle \frac{f}{g} is continuous at \displaystyle c.
Scalar Multiple	If \displaystyle g is continuous, then \displaystyle \lambda g is continuous for any constant \displaystyle \lambda.
Reciprocal	If \displaystyle g is continuous, then \displaystyle \frac{1}{g} is continuous wherever \displaystyle g(x)\ne0.
Rational Function	A rational function \displaystyle \frac{p(x)}{q(x)}, where \displaystyle p,q are polynomials, is continuous wherever \displaystyle q(x)\ne0.
Composition	If \displaystyle g is continuous at \displaystyle c and \displaystyle f is continuous at \displaystyle g(c), then \displaystyle f\circ g is continuous at \displaystyle c.
Piecewise Function	At a joining point \displaystyle x=c, check \displaystyle \lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c) for continuity.
Greatest Integer Function	The function \displaystyle f(x)=[x] is continuous at non-integer points but discontinuous at every integer.
Domain Exclusions	A function may be continuous at every point where it is defined, yet be undefined at certain points. For example, \displaystyle \tan x is continuous wherever \displaystyle \cos x\ne0

Maths Better Tip… Remember: algebraic combinations of continuous functions remain continuous wherever the resulting function is defined. For piecewise functions, always check the joining point separately using LHL = RHL = f(c).

Table 3: Continuity and Differentiability QRT – Differentiability Basics

Concept	Formula / Result
Derivative	The derivative of \displaystyle f at a point \displaystyle x is \displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}, provided the limit exists.
Derivative at a Point	At \displaystyle x=c, \displaystyle f'(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}, provided the limit exists.
Right-Hand Derivative	\displaystyle R.H.D.=\lim_{h\to0^+}\frac{f(c+h)-f(c)}{h}
Left-Hand Derivative	\displaystyle L.H.D.=\lim_{h\to0}\frac{f(c-h)-f(c)}{-h}
Differentiability at a Point	A function is differentiable at \displaystyle c if both LHD and RHD are finite and equal.
Differentiability in an Interval	A function is differentiable in \displaystyle [a,b] if it is differentiable at every point of the interval, using the right-hand derivative at \displaystyle a and left-hand derivative at \displaystyle b.
Differentiable ⇒ Continuous	If \displaystyle f is differentiable at \displaystyle c, then \displaystyle f is continuous at \displaystyle c.
Converse is False	A continuous function need not be differentiable. For example, \displaystyle f(x)=|x| is continuous everywhere but is not differentiable at \displaystyle x=0.
Corner / Sharp Point	If the left-hand and right-hand derivatives at a point are unequal, the function is not differentiable there.
Non-Differentiability	A function is not differentiable at \displaystyle x=c if the left-hand and right-hand derivatives are unequal, or if the derivative does not exist as a finite value.
Graphical Check	Graphically, a sharp corner or cusp at a point is an indication that the function may not be differentiable there.
Derivative Not Defined	If the defining derivative limit does not exist or is not finite, the function is not differentiable at that point.

Maths Better Tip… Remember: differentiability is a stronger condition than continuity. If a function is differentiable at a point, it must be continuous there; but continuity alone does not guarantee differentiability.

QRT 4: Differentiation Rules & Standard Derivatives

Concept	Formula / Result
Constant	If \displaystyle y=c, then \displaystyle \frac{dy}{dx}=0
Power Rule	\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}
Constant Multiple	\displaystyle \frac{d}{dx}\bigl[cf(x)\bigr]=c\,f'(x)
Sum / Difference	\displaystyle (u\pm v)'=u'\pm v'
Product Rule	\displaystyle (uv)'=uv'+vu' Remember: first × derivative of second + second × derivative of first
Quotient Rule	\displaystyle \left(\frac{u}{v}\right)'=\frac{vu'-uv'}{v^2} Remember: \displaystyle \frac{N}{D}\Rightarrow\frac{DN'-ND'}{D^2}
Exponential	\displaystyle \frac{d}{dx}(e^x)=e^x
General Exponential	\displaystyle \frac{d}{dx}(a^x)=a^x\log a, where \displaystyle a\gt0,\ a\ne1
Logarithmic	\displaystyle \frac{d}{dx}(\log x)=\frac{1}{x}, for \displaystyle x\gt0
Sine Cosine }	\displaystyle \frac{d}{dx}(\sin x)=\cos x
	\displaystyle \frac{d}{dx}(\cos x)=-\sin x
Tangent Cotangent }	\displaystyle \frac{d}{dx}(\tan x)=\sec^2x
	\displaystyle \frac{d}{dx}(\cot x)=-\cosec^2x
Secant Cosecant }	\displaystyle \frac{d}{dx}(\sec x)=\sec x\tan x
	\displaystyle \frac{d}{dx}(\cosec x)=-\cosec x\cot x

Quick Recall: Remember the derivatives in pairs: \displaystyle \sin\leftrightarrow\cos, \displaystyle \tan\leftrightarrow\cot, and \displaystyle \sec\leftrightarrow\cosec. The derivatives of all the co-functions — \displaystyle \cos,\cot,\cosec — carry a negative sign.

Maths Better Tip… Remember: in a product, differentiate the first factor and keep the second unchanged, then add the first unchanged times the derivative of the second. For a quotient, use “denominator × derivative of numerator − numerator × derivative of denominator”, all divided by denominator squared.

Table 5: Continuity and Differentiability QRT – Chain Rule & Implicit Differentiation

Concept	Formula / Result
Chain Rule	If \displaystyle y=f(u) and \displaystyle u=g(x), then \displaystyle \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}
Composite Function	If \displaystyle y=f(g(x)), then \displaystyle \frac{dy}{dx}=f'(g(x))\,g'(x)
Three Functions	If \displaystyle y=f(u),\ u=g(v),\ v=h(x), then \displaystyle \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}
Multiple Layers	Start with the outermost layer, move inwards, and multiply each derivative until the last derivative is 1.
Power of a Function	\displaystyle \frac{d}{dx}(f^n)=nf^{n-1}f'
Exponential of a Function	\displaystyle \frac{d}{dx}(e^f)=e^f f'
Logarithm of a Function	\displaystyle \frac{d}{dx}(\log f)=\frac{f'}{f}, for \displaystyle f\gt0
Implicit Function	When \displaystyle x and \displaystyle y are related by an equation, \displaystyle y may be treated as a function of \displaystyle x and differentiated implicitly.
Implicit Differentiation	Differentiate both sides with respect to \displaystyle x, treating \displaystyle y as a function of \displaystyle x.
Derivative of y	While differentiating with respect to \displaystyle x, \displaystyle \frac{d}{dx}(y)=\frac{dy}{dx}
Derivative of yⁿ	\displaystyle \frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}
Finding dy/dx	After differentiating an implicit equation, collect the terms containing \displaystyle \frac{dy}{dx} and solve for \displaystyle \frac{dy}{dx}

Maths Better Tip… Remember: Chain Rule = “differentiate the outer function × differentiate the inner function.” In implicit differentiation, every term involving \displaystyle y must be differentiated with respect to \displaystyle x, so \displaystyle \frac{dy}{dx} appears naturally.

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QRT 6: Derivatives of Inverse Trigonometric Functions

Function	Derivative
\displaystyle \sin^{-1}x \displaystyle \cos^{-1}x }	\displaystyle \frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}, for \displaystyle -1\lt x\lt1
	\displaystyle \frac{d}{dx}(\cos^{-1}x)=-\frac{1}{\sqrt{1-x^2}}, for \displaystyle -1\lt x\lt1
\displaystyle \tan^{-1}x \displaystyle \cot^{-1}x }	\displaystyle \frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}, for all \displaystyle x\in\mathbb{R}
	\displaystyle \frac{d}{dx}(\cot^{-1}x)=-\frac{1}{1+x^2}, for all \displaystyle x\in\mathbb{R}
\displaystyle \sec^{-1}x \displaystyle \cosec^{-1}x }	\displaystyle \frac{d}{dx}(\sec^{-1}x)=\frac{1}{|x|\sqrt{x^2-1}}, for \displaystyle |x|\gt1
	\displaystyle \frac{d}{dx}(\cosec^{-1}x)=-\frac{1}{|x|\sqrt{x^2-1}}, for \displaystyle |x|\gt1
Chain Rule Form	If \displaystyle y=f^{-1}(u) and \displaystyle u=u(x), then \displaystyle \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}
\displaystyle \sin^{-1}u \displaystyle \cos^{-1}u }	\displaystyle \frac{d}{dx}(\sin^{-1}u)=\frac{u'}{\sqrt{1-u^2}}
	\displaystyle \frac{d}{dx}(\cos^{-1}u)=-\frac{u'}{\sqrt{1-u^2}}
\displaystyle \tan^{-1}u \displaystyle \cot^{-1}u }	\displaystyle \frac{d}{dx}(\tan^{-1}u)=\frac{u'}{1+u^2}
	\displaystyle \frac{d}{dx}(\cot^{-1}u)=-\frac{u'}{1+u^2}
\displaystyle \sec^{-1}u \displaystyle \cosec^{-1}u }	\displaystyle \frac{d}{dx}(\sec^{-1}u)=\frac{u'}{|u|\sqrt{u^2-1}}
	\displaystyle \frac{d}{dx}(\cosec^{-1}u)=-\frac{u'}{|u|\sqrt{u^2-1}}

Maths Better Tip… Remember the pattern: first find the derivative of the inverse-trigonometric function with respect to its argument, then multiply by the derivative of that argument. Also watch the domain carefully, especially for \displaystyle \sin^{-1}x,\cos^{-1}x,\sec^{-1}x and \displaystyle \cosec^{-1}x.

QRT 7: Exponential & Logarithmic Functions

Concept	Formula / Result
Exponential Function	\displaystyle f(x)=a^x, where \displaystyle a\gt0,\ a\ne1
Natural Exponential	The exponential function with base \displaystyle e is \displaystyle f(x)=e^x
Derivative of \displaystyle e^x	\displaystyle \frac{d}{dx}(e^x)=e^x
Derivative of \displaystyle a^x	\displaystyle \frac{d}{dx}(a^x)=a^x\log a, where \displaystyle a\gt0,\ a\ne1
Logarithmic Function	The logarithmic function with base \displaystyle a is \displaystyle f(x)=\log_a x, where \displaystyle a\gt0,\ a\ne1,\ x\gt0
Natural Logarithm	The log function with base \displaystyle e: \displaystyle \log_e x=\log x=\ln x, for \displaystyle x\gt0
Logarithm as Inverse	\displaystyle y=\log_a x\iff a^y=x, where \displaystyle a\gt0,\ a\ne1
Derivative of \displaystyle \log x	\displaystyle \frac{d}{dx}(\log x)=\frac{1}{x}, for \displaystyle x\gt0
Derivative of \displaystyle \log_a x	\displaystyle \frac{d}{dx}(\log_a x)=\frac{1}{x\log a}, where \displaystyle a\gt0,\ a\ne1
Exponential Chain Rule	\displaystyle \frac{d}{dx}\left(e^{f(x)}\right)=e^{f(x)}f'(x)
Logarithmic Chain Rule	\displaystyle \frac{d}{dx}\left(\log f(x)\right)=\frac{f'(x)}{f(x)}, wherever \displaystyle f(x)\gt0
Change of Base	\displaystyle \log_a x=\frac{\log x}{\log a}, where \displaystyle a\gt0,\ a\ne1,\ x\gt0
Important Log Laws	\displaystyle \log(ab)=\log a+\log b; \displaystyle \log\left(\frac{a}{b}\right)=\log a-\log b; \displaystyle \log(a^n)=n\log a

Maths Better Tip… Remember: \displaystyle e^x differentiates to itself, while \displaystyle a^x gives an extra factor of \displaystyle \log a. For logarithms, always remember the domain condition \displaystyle x\gt0 before applying the standard derivative.

Table 8: Continuity and Differentiability QRT – Logarithmic Differentiation

Concept	Formula / Result
When to Use	Logarithmic differentiation is useful when the function contains products, quotients or variable powers that are difficult to differentiate directly.
Basic Step	Put \displaystyle y=f(x), take logarithm on both sides, simplify using logarithm laws, then differentiate.
Key Transformation	If \displaystyle y=f(x), then \displaystyle \log y=\log f(x)
Differentiate \displaystyle \log y	\displaystyle \frac{d}{dx}(\log y)=\frac{1}{y}\frac{dy}{dx}
Variable Power	If \displaystyle y=[f(x)]^{g(x)}, then \displaystyle \log y=g(x)\log f(x) before differentiating.
General Result (Chain + Product)	For \displaystyle y=[f(x)]^{g(x)}, \displaystyle \frac{y'}{y}=g'(x)\log f(x)+g(x)\frac{f'(x)}{f(x)}
Power Function	If \displaystyle y=x^n, logarithmic differentiation gives \displaystyle \frac{y'}{y}=\frac{n}{x}, hence \displaystyle y'=nx^{n-1}
Product Form	For \displaystyle y=uv, \displaystyle \log y=\log u+\log v, so \displaystyle \frac{y'}{y}=\frac{u'}{u}+\frac{v'}{v}
Quotient Form	For \displaystyle y=\frac{u}{v}, \displaystyle \log y=\log u-\log v, so \displaystyle \frac{y'}{y}=\frac{u'}{u}-\frac{v'}{v}
Example: \displaystyle x^x	If \displaystyle y=x^x, then \displaystyle \log y=x\log x, giving \displaystyle y'=x^x(1+\log x), for \displaystyle x\gt0
Final Step	After finding \displaystyle \frac{y'}{y}, multiply by \displaystyle y to obtain \displaystyle y'

Maths Better Tip… Remember the flow: take log → expand using log laws → differentiate → find \displaystyle \frac{y'}{y} → multiply by \displaystyle y. It is especially powerful when both the base and exponent contain \displaystyle x.

Table 9: Continuity and Differentiability QRT – Parametric Differentiation

Concept	Formula / Result
Parametric Form	If \displaystyle x=f(t) and \displaystyle y=g(t), then both \displaystyle x and \displaystyle y are expressed in terms of the parameter \displaystyle t
First Derivative	\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}, provided \displaystyle \frac{dx}{dt}\ne0
Equivalent Form	\displaystyle \frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}
Step 1	Differentiate \displaystyle y=g(t) with respect to \displaystyle t to find \displaystyle \frac{dy}{dt}
Step 2	Differentiate \displaystyle x=f(t) with respect to \displaystyle t to find \displaystyle \frac{dx}{dt}
Step 3	Divide \displaystyle \frac{dy}{dt} by \displaystyle \frac{dx}{dt} to obtain \displaystyle \frac{dy}{dx}
Second Derivative	\displaystyle \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}
Alternative Form	\displaystyle \frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dt}\left(\frac{dy}{dx}\right)\frac{dt}{dx}
Caution	For the second derivative, do not simply divide \displaystyle \frac{d^2y}{dt^2} by \displaystyle \frac{d^2x}{dt^2}. First differentiate \displaystyle \frac{dy}{dx} with respect to \displaystyle t, then divide by \displaystyle \frac{dx}{dt}

Maths Better Tip… Remember: in parametric differentiation, first differentiate both \displaystyle x and \displaystyle y with respect to \displaystyle t, then divide. For the second derivative, differentiate \displaystyle \frac{dy}{dx} with respect to \displaystyle t and divide once again by \displaystyle \frac{dx}{dt}.

Table 10: Continuity and Differentiability QRT – Second Order & Higher Derivatives

Concept	Formula / Result
Second Derivative	If \displaystyle y=f(x), the second derivative is \displaystyle \frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)
Common Notations	The second derivative may be written as \displaystyle f''(x),\quad y'',\quad \frac{d^2y}{dx^2},\quad D^2y
Successive Differentiation	First find \displaystyle y', then differentiate \displaystyle y' again with respect to \displaystyle x to obtain \displaystyle y''
Third Derivative	\displaystyle \frac{d^3y}{dx^3}=\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)
nth Derivative	The \displaystyle nth derivative is denoted by \displaystyle \frac{d^ny}{dx^n} or \displaystyle f^{(n)}(x)
Product: First Derivative	If \displaystyle y=uv, then \displaystyle y'=uv'+vu'
Product: Second Derivative	If \displaystyle y=uv, then \displaystyle y''=uv''+2u'v'+u''v
Parametric Second Derivative	If \displaystyle x=f(t),\ y=g(t), then \displaystyle \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}
Important Condition	For the parametric second derivative, \displaystyle \frac{dx}{dt}\ne0 is required at the point under consideration.
Constant Second Derivative	If \displaystyle y=ax^2+bx+c, then \displaystyle \frac{d^2y}{dx^2}=2a
Linear Function	If \displaystyle y=ax+b, then \displaystyle \frac{d^2y}{dx^2}=0
Higher Derivatives	Higher derivatives are obtained successively: \displaystyle y',\ y'',\ y''',\ldots, by differentiating the preceding derivative.

Maths Better Tip… Remember: the second derivative is simply the derivative of the first derivative. In parametric form, differentiate \displaystyle \frac{dy}{dx} with respect to \displaystyle t and divide by \displaystyle \frac{dx}{dt}.

Quick Questions — Test Yourself

Maths Better Tip… Try to answer each question before opening it. Then click the arrow to reveal the answer and check yourself.

What are the three conditions for \displaystyle f(x) to be continuous at \displaystyle x=c?

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\displaystyle f(c) exists, \displaystyle \lim_{x\to c}f(x) exists, and \displaystyle \lim_{x\to c}f(x)=f(c)

If a function is continuous at \displaystyle x=c, is it necessarily differentiable there?

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No. Continuity does not necessarily imply differentiability. For example, \displaystyle f(x)=|x| is continuous at \displaystyle x=0 but is not differentiable there.

If a function is differentiable at \displaystyle x=c, what can be concluded about its continuity there?

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Differentiability at \displaystyle x=c implies continuity at \displaystyle x=c

For differentiability at \displaystyle x=c, what must be true about the LHD and RHD?

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Both must exist as finite values and must be equal: \displaystyle f'_-(c)=f'_+(c)

Differentiate \displaystyle y=\sin(x^2)

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\displaystyle \frac{dy}{dx}=2x\cos(x^2)

Differentiate \displaystyle y=\sin^{-1}(3x)

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\displaystyle \frac{dy}{dx}=\frac{3}{\sqrt{1-9x^2}}

Differentiate \displaystyle y=e^{\cos x}

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\displaystyle \frac{dy}{dx}=-e^{\cos x}\sin x

Find \displaystyle \frac{dy}{dx} if \displaystyle y+\sin y=\cos x

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\displaystyle \frac{dy}{dx}=-\frac{\sin x}{1+\cos y}

What is the derivative of \displaystyle \log x?

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\displaystyle \frac{d}{dx}(\log x)=\frac{1}{x}, for \displaystyle x\gt0

Using logarithmic differentiation, find the derivative of \displaystyle y=x^x, for \displaystyle x\gt0

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\displaystyle y'=x^x(1+\log x)

If \displaystyle x=a\cos t and \displaystyle y=a\sin t, find \displaystyle \frac{dy}{dx}

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\displaystyle \frac{dy}{dx}=-\cot t

If \displaystyle y=ax^2+bx+c, what is \displaystyle \frac{d^2y}{dx^2}?

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\displaystyle \frac{d^2y}{dx^2}=2a

For parametric equations \displaystyle x=f(t),\ y=g(t), how is \displaystyle \frac{d^2y}{dx^2} found?

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\displaystyle \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}, provided \displaystyle \frac{dx}{dt}\ne0

Is \displaystyle e^{\log x}=x true for every real \displaystyle x?

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No. Since \displaystyle \log x is defined only for \displaystyle x\gt0, the identity holds only for positive \displaystyle x

What is the main difference between continuity and differentiability at a point?

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Continuity requires the function value and limit to agree, while differentiability requires the derivative limit to exist. Differentiability implies continuity, but continuity does not imply differentiability.

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