NCERT Solutions for Class 12 Mathematics Chapter 5: Continuity and Differentiability – Free PDF Download

Complete, verified solutions to the NCERT Class 12 Maths Chapter 5 (Continuity and Differentiability) exercise questions. These Class 12 Mathematics Chapter 5 solutions are also useful as quick revision notes before exams.

NCERT Exercise Solutions

5.1 Prove that the function f(x)=5x-3 is continuous at x=0, at x=-3 and at x=5.
Ans: f is a polynomial, hence continuous everywhere. At x=0: lim f(x)=f(0)=-3. At x=-3: lim f(x)=f(-3)=-18. At x=5: lim f(x)=f(5)=22. In each case LHL=RHL=f(c), so f is continuous at all three points.

5.2 Examine the continuity of the function f(x)=2x²-1 at x=3.
Ans: f(3)=2(9)-1=17. lim(x→3) f(x)=2(9)-1=17. Since lim=f(3), f is continuous at x=3.

5.3 Find all points of discontinuity of f, where f(x) = 2x+3 if x≤2, 2x-3 if x>2.
Ans: At x=2: LHL=2(2)+3=7, RHL=2(2)-3=1. Since LHL≠RHL, f is discontinuous at x=2. Continuous everywhere else (both pieces are polynomials).

5.4 Discuss the continuity of the function f(x)=xⁿ (n a positive integer).
Ans: f(x)=xⁿ is a polynomial function, and every polynomial function is continuous at every real number, so f is continuous everywhere.

5.5 Is the function defined by f(x)=|x| a continuous function?
Ans: Yes. f(x)=x for x≥0 and f(x)=-x for x<0. At x=0: LHL=lim(-x)=0, RHL=lim(x)=0, f(0)=0. All equal, so continuous at 0, and continuous elsewhere since it is piecewise polynomial. So |x| is continuous for all real x.

Differentiation Rules Used Throughout the Chapter

Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x). Derivative of e᷀: d/dx(e᷀)=e᷀. Derivative of log x: d/dx(log x)=1/x. Implicit differentiation: differentiate both sides w.r.t. x, treating y as a function of x, then solve for dy/dx. Logarithmic differentiation: used when the function has variable base AND variable exponent (like x᷀) — take log of both sides first.

Frequently Asked Questions

What is the relationship between differentiability and continuity?
Every differentiable function is continuous, but not every continuous function is differentiable (e.g. |x| is continuous but not differentiable at x=0).

When do we use logarithmic differentiation?
When the function has the form [f(x)]^g(x), i.e. both variable base and variable exponent.

Written by Satish

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