Another Mean Value Theorem problem, just basically asking for confirmation. (Problem below)

Question

mendicant_bias · Answer

Does the following function satisfy the MVT on the given interval? Explain why or why not.

$$f(x) = \left\{ \frac{ \sin(x) }{ x }, -\pi \le x < 0 \right\}$$$$f(x) = \left\{ 0, x = 0 \right\}$$
$$f'(x) = \frac{ xcox(x) - \sin(x) }{ x ^{2} }$$$$f'(x) = 0$$
My answer: No, it does not, because there is a discontinuity between the point where the function accounting for the interior (first one) connects with the endpoint 0 where x = 0. Despite the interior being entirely differentiable, it is discontinuous exactly where it connects with an endpoint, and Rolle's Theorem fails, which is a necessary assumption for the Mean Value Theorem to work.