(Solved): Let \( I:=[a, b] \) and let \( f: I \rightarrow \mathbb{R} \) be differentiab ...

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Let \( I:=[a, b] \) and let \( f: I \rightarrow \mathbb{R} \) be differentiable at \( c \in I \). Show that for every \( \varepsilon>0 \) there exists \( \delta>0 \) such that if \( 0<|x-y|<\delta \) and \( a \leq x \leq c \leq y \leq b \), then \[ \left|\frac{f(x)-f(y)}{x-y}-f^{\prime}(c)\right|<\varepsilon \]