If $f:(a,b)\to \mathbb{R}$ is differentiable in a point $x\in (a,b)$, show that $\lim\limits_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}$ exists
137527
if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$
105527
Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$
17646
If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$
53236
Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$
111470
Proof of differentiability of $f(x)$
135693
Suppose f is a twice differentiable function with $f(0)=0$ and $f(1)=1$, $f'(0)=f'(1)=0$. Then show that $|f''(x)| > 4$ for some $x$ in $(0,1)$
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$
49781
Existence of the absolute value of the limit implies that either $f \ $ or $\bar{f} \ $ is complex-differentiable
49511
Do we have $\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$?
41037
Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
22686
Show $\lim_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
120784
Differentiablility over closed intervals
119053
How to prove that $\displaystyle f''(a)=\lim_{h \to 0}\frac{f(a+2h)-2f(a+h)+f(a)}{h^2}$
142490
Alternative proof for differentiability of inverse function?
136980
Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
33130
$f$ differentiable except in $1$ point