f is continuous at x = a if:

lim(x→a) f(x) = f(a). The limit must exist and equal the function value.

A polynomial function is continuous:

For all real x. Polynomials have no breaks.

f(x) = (x² − 9)/(x − 3) for x ≠ 3 and f(3) = 5. At x = 3, f has:

A removable discontinuity. The limit is 6, which exists but is not equal to f(3).

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 8: Continuity

Short notes, key concepts, quick revision and practice MCQs with answers

A function is continuous if its graph has no break, jump or gap. The chapter defines continuity at a point using limits, continuity from the left and the right, continuity over open and closed intervals, types of discontinuity, and uses these ideas to find unknown constants that make a function continuous.

Key concepts: Continuity

Continuity at a point: f is continuous at x = a if (1) f(a) is defined, (2) lim(x→a) f(x) exists and (3) lim(x→a) f(x) = f(a).
Continuity from right and left: f is continuous from the right at a if lim(x→a⁺) f(x) = f(a), and from the left at b if lim(x→b⁻) f(x) = f(b).
Continuity over an interval: Over an open interval: continuous at every point of it. Over a closed interval [a, b]: continuous on (a, b), from the right at a and from the left at b.
Removable discontinuity: The limit exists but f(a) is not defined or is not equal to the limit. Redefining f(a) as the limit removes the discontinuity.
Jump (irremovable) discontinuity: The left hand and right hand limits exist but are unequal, so the limit does not exist.
Continuous functions: Polynomials, eˣ and aˣ are continuous for all real x; log x is continuous for x > 0; a rational function is continuous wherever its denominator is not zero. Sums, differences, products and quotients (non-zero denominator) of continuous functions are continuous.
Finding unknown constants: For a piecewise function, equate the limit (or both one-sided limits) at the joining point with the function value and solve for the constant.

Quick revision: important points

Continuity at a needs all three conditions; failing any one gives a discontinuity.
|x| is continuous everywhere, including x = 0.
The greatest integer function [x] is discontinuous at every integer.
For (xⁿ − aⁿ)/(x − a) type functions, the value that makes f continuous at a is n aⁿ⁻¹.
For (eˣ − 1)/x or log(1 + x)/x, the value at 0 must be 1 for continuity.

Practice MCQs with answers

f is continuous at x = a if:
1. f(a) = 0
2. lim(x→a) f(x) = f(a)
3. f is defined near a
4. f′(a) exists
Show answer
Answer: B. lim(x→a) f(x) = f(a)
The limit must exist and equal the function value.
A polynomial function is continuous:
1. Only at x = 0
2. Only for x > 0
3. For all real x
4. Only at integers
Show answer
Answer: C. For all real x
Polynomials have no breaks.
f(x) = 1/(x − 2) is discontinuous at:
1. x = 0
2. x = 2
3. x = −2
4. Nowhere
Show answer
Answer: B. x = 2
f(2) is not defined.
If f(x) = (x² − 4)/(x − 2) for x ≠ 2 and f(2) = k is continuous at x = 2, then k is:
1. 2
2. 0
3. 4
4. −4
Show answer
Answer: C. 4
The limit is 2 + 2 = 4.
f(x) = (x² − 9)/(x − 3) for x ≠ 3 and f(3) = 5. At x = 3, f has:
1. Continuity
2. A removable discontinuity
3. A jump discontinuity
4. An infinite discontinuity
Show answer
Answer: B. A removable discontinuity
The limit is 6, which exists but is not equal to f(3).

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters