Answer: Continuous but not differentiable.
- A Differentiable but not continuous
- B Continuous and differentiable
- C Continuous but not differentiable
- D Neither continuous nor differentiable
Correct answer: C. Continuous but not differentiable
Explanation: f(x) = |x| is continuous everywhere, but at x = 0 the left-hand derivative is -1 and the right-hand derivative is +1, so it is not differentiable there.
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.