If f is differentiable on (a,b) then f is ______ on [a,b].

If f is differentiable on (a,b) then f is ______ on [a,b].

• A. Continuous
• B. Differentiable
• C. Bounded
• D. Integrable

Answer: (A)

Differentiability implies continuity. If f has a derivative at every point inside (a,b), then for each such point c, the limit of [f(x) − f(c)]/(x − c) as x approaches c exists and equals f′(c). That forces lim x→c f(x) = f(c), so f is continuous at every interior point. Since the function is defined on the closed interval [a,b], the usual way we describe its behavior on that interval is that it is continuous there as well, with the understanding that endpoints involve one‑sided behavior. The differentiability on the open interval guarantees no jumps or breaks inside, which is the essential takeaway.

Differentiability implies continuity. If f has a derivative at every point inside (a,b), then for each such point c, the limit of [f(x) − f(c)]/(x − c) as x approaches c exists and equals f′(c). That forces lim x→c f(x) = f(c), so f is continuous at every interior point. Since the function is defined on the closed interval [a,b], the usual way we describe its behavior on that interval is that it is continuous there as well, with the understanding that endpoints involve one‑sided behavior. The differentiability on the open interval guarantees no jumps or breaks inside, which is the essential takeaway.