Continuity and Differentiability for CUET Maths

Q: When is a function continuous at a point?

f is continuous at x = a when three numbers agree: the left-hand limit, the right-hand limit and the value f(a).

Q: How do you find the constant that makes a function continuous?

Write the left-hand limit, the right-hand limit and the value at the joining point in terms of the constant, and set them equal. For a formula of the form 0/0 at the point, the constant must equal the limit.

Q: Which functions are continuous?

Polynomials, sin x, cos x, e^x and x are continuous everywhere; log x, tan x and rational functions are continuous at every point of their domain. Sums, differences, products and compositions of continuous functions are continuous, and so is a quotient wherever the denominator is not 0. The greatest integer function [x] jumps at every integer.

Q: What makes a function not differentiable at a point?

f is differentiable at a when the left-hand and right-hand derivatives agree. A break, a jump, or a corner where two different slopes meet spoils it. A function that is differentiable at a point is continuous there, but not the other way round: x is continuous at 0 with a corner.

About 5 min to learn · 125 questions

Continuity and differentiability open the calculus of CUET Maths Section B1. The questions are short and exact: is a function continuous at a point, which constant makes it continuous, where is it not differentiable. Learn each rule here, from testing a function at a point and finding the constant to the algebra of continuous functions, corners and differentiability, with worked examples, time-saving shortcuts and the traps CUET sets, then practise on free questions in CUET's own formats, each fully solved.

Syllabus area: Section B1: Mathematics

On this page · 10 sections

01What continuity means
02Continuity at a point
03Choosing a constant for continuity
04Continuous functions
05Differentiability
06How CUET asks this chapter
07Exam shortcuts
08Common mistakes
09Quick revision
10Check your understanding

What continuity means

A parking lot charges ₹20 for the first two hours and then ₹15 for every further hour or part of one. Plot the charge against time and the graph jumps at 2 hours, 3 hours, 4 hours. A function is continuous at a point when its graph has no such break there: you could draw it through that point without lifting the pen. This chapter turns that picture into a test you can run on a formula.

When is a function continuous at a point?

$f$ is continuous at $x = a$ when three numbers agree: the left-hand limit, the right-hand limit and the value $f(a)$ .

\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = f(a)

Continuity at a point

If the two one-sided limits differ, the graph jumps. If they agree but differ from $f(a)$ , the limit exists but the function has been given the wrong value there. Either way, it is not continuous.

Example 1 · A piecewise function at its join

Is $f(x) = 2x + 3$ for $x < 1$ and $f(x) = 6 - x$ for $x \ge 1$ continuous at $x = 1$ ?

Left-hand limit: $2(1) + 3 = 5$ .
Right-hand limit and value: $6 - 1 = 5$ .

Answer

All three are 5, so $f$ is continuous at $x = 1$ .

Choosing a constant for continuity

How do you find the constant that makes a function continuous?

Write the left-hand limit, the right-hand limit and the value at the joining point in terms of the constant, and set them equal. For a formula of the form $\frac{0}{0}$ at the point, the constant must equal the limit.

Example 2 · Joining two pieces

For which $k$ is $f(x) = kx + 5$ for $x \le 2$ and $f(x) = x^{2} - 1$ for $x > 2$ continuous at $x = 2$ ?

The pieces must meet: $2k + 5 = 2^{2} - 1 = 3$ .
$2k = -2$ .

Answer

k = -1

Example 3 · A constant from a limit

$f(x) = \frac{1 - \cos 6x}{x^{2}}$ for $x \ne 0$ and $f(0) = k$ . Find $k$ for continuity at 0.

$1 - \cos 6x = 2\sin^{2} 3x$ , so $f(x) = 2\left(\frac{\sin 3x}{x}\right)^{2}$ .
$\frac{\sin 3x}{x} \to 3$ , so the limit is $2 \times 9 = 18$ .

Answer

k = 18

Key idea

Limits to keep ready: $\frac{\sin ax}{bx} \to \frac{a}{b}$ , $\frac{e^{ax} - 1}{x} \to a$ , $\frac{1 - \cos ax}{x^{2}} \to \frac{a^{2}}{2}$ , and $\frac{x^{n} - a^{n}}{x - a} \to na^{n - 1}$ .

Continuous functions

Which functions are continuous?

Polynomials, $\sin x$ , $\cos x$ , $e^{x}$ and $\lvert x \rvert$ are continuous everywhere; $\log x$ , $\tan x$ and rational functions are continuous at every point of their domain. Sums, differences, products and compositions of continuous functions are continuous, and so is a quotient wherever the denominator is not 0. The greatest integer function $[x]$ jumps at every integer.

Watch out

Continuity is checked only at points of the domain. $\frac{1}{x}$ is a continuous function: 0 is not in its domain, so it is not a break of the function.

Differentiability

What makes a function not differentiable at a point?

$f$ is differentiable at $a$ when the left-hand and right-hand derivatives agree. A break, a jump, or a corner where two different slopes meet spoils it. A function that is differentiable at a point is continuous there, but not the other way round: $\lvert x \rvert$ is continuous at 0 with a corner.

Example 4 · A smooth join

Is $f(x) = x^{2}$ for $x \le 2$ and $f(x) = 4x - 4$ for $x > 2$ differentiable at $x = 2$ ?

Values: $2^{2} = 4$ and $4(2) - 4 = 4$ : continuous.
Slopes: $2x = 4$ on the left and 4 on the right: equal.

Answer

Differentiable at $x = 2$ , with derivative 4.

Example 5 · Counting corners

At how many points is $f(x) = \lvert x + 2 \rvert + \lvert x - 5 \rvert$ not differentiable?

Each modulus has a corner where its inside is 0: $x = -2$ and $x = 5$ .
The corners are at different points, so neither cancels the other.

Answer

2 points

How CUET asks this chapter

At a point: whether a piecewise or $\frac{0}{0}$ function is continuous, or the value of a one-sided limit.
Constants: the $k$ that makes a function continuous, or the $a$ and $b$ that make it continuous or differentiable.
Continuous functions: which functions are continuous everywhere, the algebra of continuous functions, points of discontinuity of $[x]$ .
Differentiability: corners of $\lvert x \rvert$ -type functions, one-sided derivatives, and the link with continuity.

Exam shortcuts

Continuity at $a$ : left limit = right limit = $f(a)$ .
Differentiability at a join: the values match and the slopes match.
$\lvert x - a \rvert$ has one corner, at $a$ ; $x\lvert x \rvert$ and $\lvert x \rvert^{3}$ have none.
Differentiable ⇒ continuous; continuous does not imply differentiable.

Common mistakes

Treating a change of formula as a break even when the pieces meet.
Forgetting that a limit can exist while the value is different.
Counting points outside the domain as discontinuities.
Checking only the values at a join, not the slopes, for differentiability.

Quick revision

Continuity

LHL = RHL = $f(a)$
Constant: set the three equal

Continuous functions

Sums, products, compositions stay continuous
$[x]$ jumps at integers

Differentiability

Left and right derivatives agree
Corner: continuous, not differentiable

Next in Section B1, the rules that differentiate composite, inverse and exponential functions: chain rule, logarithmic differentiation and rate of change.

Next · 09Quick revision

Check your understanding

125 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.

Continuity at a pointQ1 of 125Q1

Continuity at a point

30 questions

List I	List II
A. $[x]$	I. 3 points
B. $[x^{2}]$	II. No point
C. $\left[x + \frac{1}{2}\right]$	III. 1 point
D. $\left[\frac{x}{2}\right]$	IV. 2 points

List I	List II
A. $\lvert x \rvert$	I. $-2$ and $2$
B. $x + \lvert x \rvert$	II. $-3$ and $-1$
C. $\lvert x \rvert - 2x$	III. $0$ and $2$
D. $\lvert \sin 2x \rvert$	IV. $-1$ and $1$

List I	List II
A. $x^{2}$	I. Neither continuous nor differentiable
B. $\lvert x \rvert$	II. Differentiable, with derivative 0
C. $\frac{\lvert x \rvert}{x}$ with value 0 at 0	III. Differentiable, though built from $\lvert x \rvert$
D. $\lvert x \rvert^{3}$	IV. Continuous but not differentiable

Check your understanding

Continuity at a point

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

Question 22

Question 23

Question 24

Question 25

Question 26

Question 27

Question 28

Question 29

Question 30

Choosing a constant for continuity

Question 31

Question 32

Question 33

Question 34

Question 35

Question 36

Question 37

Question 38

Question 39

Question 40

Question 41

Question 42

Question 43

Question 44

Question 45

Question 46

Question 47

Question 48

Question 49

Question 50

Question 51

Question 52

Question 53

Question 54

Question 55

Question 56

Question 57

Continuous functions

Question 58

Question 59

Question 60

Question 61

Question 62

Question 63

Question 64

Question 65

Question 66

Question 67

Question 68

Question 69

Question 70

Question 71

Question 72

Question 73

Question 74

Question 75

Question 76