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The hypotheses ensure that the function behaves smoothly enough for an interior tangent to match the secant slope. A sharp interior corner or a jump at an endpoint can remove this guarantee.

Conditions: The function fails to be continuous on $[a,b]$ or differentiable on $(a,b)$.

The hypotheses ensure that the function behaves smoothly enough for an interior tangent to match the secant slope. A sharp interior corner or a jump at an endpoint can remove this guarantee.

Conditions: The function fails to be continuous on $[a,b]$ or differentiable on $(a,b)$.

No, the point $c$ is not necessarily unique. The theorem guarantees the existence of at least one such point in the open interval $(a,b)$, but there may be multiple points where the instantaneous rate of change equals the average rate of change.

Conditions: $f$ is continuous on $[a,b]$; $f$ is differentiable on $(a,b)$

No, the point $c$ is not necessarily unique. The theorem guarantees the existence of at least one such point in the open interval $(a,b)$, but there may be multiple points where the instantaneous rate of change equals the average rate of change.

Conditions: $f$ is continuous on $[a,b]$; $f$ is differentiable on $(a,b)$