Answer: f is differentiable at x=0 with f'(0)=0.
- A f is not continuous at x=0
- B f is continuous but not differentiable at x=0
- C f is differentiable at x=0 with f'(0)=0
- D f is differentiable at x=0 with f'(0)=1
Correct answer: C. f is differentiable at x=0 with f'(0)=0
Explanation: f(x) = x<sup>2</sup> for x>=0 and f(x) = -x<sup>2</sup> for x<0. Both one-sided derivatives at 0 equal 0, so f is differentiable at x=0 with f'(0)=0.
f(x)=|x| is perfectly continuous at x=0 (no break, no jump), but the curve has a sharp corner there: approaching from the left gives slope -1 and from the right gives slope +1 - since these one-sided derivatives disagree, the function is not differentiable at that point.
Concept context
When a function has no breaks (continuity) and when it has a well-defined slope (differentiability), plus rules for differentiating implicit, inverse trig, exponential, log, and parametric functions.