Example 1 · level 1
f(x) = kx − 1 for x ≤ 1, and f(x) = x² − 3 for x > 1. Find k so that f is continuous at x = 1.
Show solution
- LHL = f(1) = 1k − 1.
- RHL = −2.
- 1k − 1 = −2 → k = −1.
Answer: -1
CLASS 12 MATHS · CONTINUITY AND DIFFERENTIABILITY
Continuity and differentiability
f is continuous at c when the left limit, right limit and f(c) are all equal. A sharp corner (like |x| at 0) is continuous but not differentiable.
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Worked examples
Example 2 · level 2
f(x) = sin(4x)/x for x ≠ 0, and f(0) = k. Find k so that f is continuous at x = 0.
Show solution
- lim (x→0) sin(4x)/x = 4 × lim sin(4x)/(4x) = 4 × 1 = 4.
- So k = 4.
Answer: 4
Example 3 · level 3
Which of these functions is NOT differentiable at x = −3?
- (x + 3)²
- (x + 3)³
- |x + 3| + x²
- sin(x + 3)
Show solution
- |x + 3| + x² has a sharp corner at x = −3, so its derivative does not exist there.
- The other options are built from smooth functions (or |t| multiplied by t, whose derivative at the corner is 0), so they are differentiable.
Answer: |x + 3| + x²