Prove that if $f$ is a differentiable function and $f'(x)$ is monotone, then $f'(x)$ is continuous.
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
109896
Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.
158423
How to prove that if $f$ is continuous a.e., then it is measurable.
151573
Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
71738
Continuously differentiable function iff $|f(x + h) - f(x + t) - l(h - t)| \leq \epsilon |h-t|$
105652
Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
13852
Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
58664
Proving a function is continuous for a fixed variable
48275
$f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
78834
if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
100740
proof of that a continuous function has a fixed point
35264
Must a monotone function have a monotone derivative?
127114
Differentiability implying continuity
53497
Prove that the function is continuous and differentiable (not as easy as it sounds imo)
76908
If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$