Limits, Continuity & Differentiability
Continuity vs Differentiability
Grade 12
Question:
<p>Which of the following is continuous everywhere in its domain but has at least one point where it is not differentiable?</p>
<p>(a) \(f(x) = x^{1/3}\)</p>
<p>(b) \(f(x) = \frac{x}{x}\)</p>
<p>(c) \(f(x) = e^x\)</p>
<p>(d) \(f(x) = \tan x\)</p>
Step-by-Step Solution
Key Concept: A function can be continuous at a point but fail to be differentiable there. The cube root function is a classic example with a vertical tangent at the origin.
<p><strong>Option (a):</strong> $f(x) = x^{1/3}$ is continuous everywhere (including at $x=0$ in its domain $\mathbb{R}$). However, $f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3x^{2/3}}$ is undefined at $x=0$, so it is not differentiable at $x=0$.</p><p><strong>Option (b):</strong> $f(x) = \frac{x}{x} = 1$ for $x \neq 0$ is not defined at $x = 0$, so its domain excludes $0$.</p><p><strong>Option (c):</strong> $f(x) = e^x$ is differentiable everywhere in its domain.</p><p><strong>Option (d):</strong> $f(x) = \tan x$ is not continuous at $x = \frac{\pi}{2} + n\pi$.</p><p>∴ Answer is (a).</p>
Correct Answer: a