Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Which function is continuous everywhere in its domain but has at least one point where it is not differentiable?</p>
<strong>f(x)=x^{1/3}</strong>
f(x)=|x|/x
f(x)=e^{-x}
f(x)=tan x
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(A) <span class="math-inline">$f(x)=x^{1/3}$</span>: continuous everywhere; not differentiable at <span class="math-inline">$x=0$</span> (vertical tangent). ✓</p><p>(B) <span class="math-inline">$f(x)=|x|/x$</span>: discontinuous at <span class="math-inline">$x=0$</span> (domain excludes 0), so not applicable.</p><p>(C) <span class="math-inline">$f(x)=e^{-x}$</span>: everywhere differentiable.</p><p>(D) <span class="math-inline">$f(x)=\tan x$</span>: discontinuous at odd multiples of <span class="math-inline">$\pi/2$</span>.</p><p><strong>Answer: (A) f(x)=x^{1/3}</strong></p><div class="key-concept"><strong>Key Concept:</strong> Continuous but non-differentiable — corner, cusp, or vertical tangent</div></div>
Correct Answer: 1