Why do mean value theorems have open interval for differentiablity while closed for continuity?
74433
Why does the Mean Value Theorem require a closed interval for continuity and an open interval for differentiability?
154572
Continuity of a function at an isolated point
141945
Why is $f(x) = \sqrt{x}$ is continuous at $x=0$? Does $\lim_{x \to 0^{-}} \sqrt{x}$ exist?
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
18082
A continuous function that attains neither its minimum nor its maximum at any open interval is monotone
142459
Continuity of a Function in Terms of Closure and Interior
49587
Does a differentiable everywhere function have a continuous derivative?
52388
The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition
12284
Mean Value Theorem Motivation
7437
Converse of mean value theorem
51546
Geometrical Interpertation of Cauchy's Mean Value Theorem
115635
Differentiability of antiderivative and continuity of derivative
111470
Proof of differentiability of $f(x)$
30629
mean value theorem question
39674
Rolle's Theorem and the Mean Value Theorem
59839
In $\epsilon$-$\delta$ continuity over an interval, $|x-c|<\delta$ implies $|f(x)-f(c)|<\epsilon$. Since both $c$ and $x$ vary, what's the difference?
129765
Mean value theorem for the second derivative, when the first derivative is zero at endpoints