Answer: lim(x->a) f(x) = f(a).
- A f(a) is defined, without checking the limit
- B lim(x->a) f(x) exists, without comparing it to f(a)
- C lim(x->a) f(x) = f(a)
- D f is differentiable at a, a stronger but different condition
Correct answer: C. lim(x->a) f(x) = f(a)
Explanation: Continuity at a point requires the limit to exist and equal the function value there: lim(x->a) f(x) = f(a).
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.