Class 12 • mathematics • Chapter 5
CONTINUITY AND DIFFERENTIABILITY
True & False Quiz
Unbroken. Smooth. Differentiable.
✓True
✗False
25
Questions
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Ch.5
Chapter
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XII
Class
Why True & False for CONTINUITY AND DIFFERENTIABILITY?
How this format sharpens your conceptual clarity
🔵 Continuity and differentiability form the bridge from limits into the calculus used throughout the rest of the syllabus.
✅ T/F probes the key hierarchy: differentiable ⇒ continuous, but continuous does NOT imply differentiable.
🎯 Trap: |x| is continuous everywhere but NOT differentiable at x = 0 — a textbook counterexample tested every year.
📋
Read each statement carefully. Click True or False — instant feedback with explanation appears. Submit anytime; unattempted questions are marked Skipped.
Q 1
If a function \(f\) is continuous at \(x=a\), then \(f(a)\) must be defined.
Q 2
If \(f\) is continuous at \(x=a\), then \(\displaystyle \lim_{x\to a}f(x)\) exists.
Q 3
Every polynomial function is continuous at every real number.
Q 4
The function \(f(x)=\dfrac{1}{x-3}\) is continuous at \(x=3\).
Q 5
If a function is differentiable at \(x=a\), then it is necessarily continuous at \(x=a\).
Q 6
If a function is continuous at \(x=a\), then it must be differentiable at \(x=a\).
Q 7
The function \(f(x)=|x-2|\) is continuous at \(x=2\).
Q 8
The function \(f(x)=|x|\) is differentiable at \(x=0\).
Q 9
If the left-hand and right-hand limits of \(f(x)\) at \(x=a\) are unequal, then \(f\) is discontinuous at \(x=a\).
Q 10
If \(\displaystyle \lim_{x\to a}f(x)=f(a)\), then \(f\) is continuous at \(x=a\).
Q 11
The function \(f(x)=\dfrac{x^2-9}{x-3}\) is continuous at \(x=3\).
Q 12
If \(f\) and \(g\) are differentiable at \(x=a\), then \(f+g\) is differentiable at \(x=a\).
Q 13
If \(f\) and \(g\) are differentiable at \(x=a\), then \(fg\) is differentiable at \(x=a\).
Q 14
If \(f\) and \(g\) are differentiable at \(x=a\), then \(\dfrac{f}{g}\) is differentiable at \(x=a\) regardless of the value of \(g(a)\).
Q 15
If \(f\) is differentiable at \(x=a\), then its left-hand derivative and right-hand derivative at \(a\) are equal.
Q 16
The function \(f(x)=x|x|\) is differentiable at \(x=0\).
Q 17
If \(f'(a)=0\), then \(f\) must have a local maximum at \(x=a\).
Q 18
If \(f\) is differentiable on an interval, then \(f\) is continuous on that interval.
Q 19
The function \(f(x)=\sin x\) is differentiable for every real \(x\), and \(f'(x)=\cos x\).
Q 20
If \(f(x)=\sin^{-1}x\), then \(f'(x)=\dfrac{1}{\sqrt{1-x^2}}\) for every real \(x\).
Q 21
If \(y=\tan^{-1}x\), then \(\dfrac{dy}{dx}=\dfrac{1}{1+x^2}\) for every \(x\in\mathbb{R}\).
Q 22
If \(y=\sin^{-1}(2x)\), then \(\dfrac{dy}{dx}=\dfrac{2}{\sqrt{1-4x^2}}\) wherever the derivative exists.
Q 23
If \(f(x)=x^x\) for \(x>0\), then \(f'(x)=x^x(1+\ln x)\).
Q 24
If \(y=(\sin x)^x\), then \(y'=(\sin x)^x\left(\ln(\sin x)+x\cot x\right)\) wherever \(\sin x>0\).
Q 25
If \(f\) is differentiable at \(x=a\) and \(f'(a)=0\), then the function must be constant in some neighbourhood of \(a\).
Key Takeaways — CONTINUITY AND DIFFERENTIABILITY
Core facts for CBSE Boards & JEE
1
Every differentiable function is continuous, but a continuous function need NOT be differentiable (e.g. |x| at x=0).
2
A function is continuous at x=a if LHL = RHL = f(a) — all three must match.
3
Logarithmic differentiation is used for functions of the form [f(x)]^g(x), taking log before differentiating.
4
d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x (for x > 0); d/dx(aˣ) = aˣ ln a.
5
Chain Rule: d/dx[f(g(x))] = f′(g(x)) · g′(x) — essential for composite and parametric functions.
6
Second order derivative d²y/dx² measures the rate of change of the slope, not the slope itself.