Prove that if $\lim_{x\rightarrow a}f'(x)$ exists and is finite for all $a \in \mathbb R$,then $f'$ is continous
1371
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
11879
If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
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 $\mathbb{R}$
34940
Continuity of $f(x) = x^n$
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
40297
Proof of uniform limit of Continuous Functions
54321
Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative
131992
A problem on continuous functions
16826
prove that every infinite subset of compact set H has a limit point (Explanation)
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
64381
Prove that if one of $\lim_{x \rightarrow 0 }{f(x)}$ and $\lim_{x \rightarrow 0}{f(x^3)}$ exists, then the other one also exists.
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
52632
$f\colon(0,\infty)\to \mathbb R$ be continuous ; $f(x)\le f(nx) , \forall n \in \mathbb N , \forall x >0$ , then $\lim_{x\to \infty} f(x)$ exists?
145428
Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
15579
Show that $f$ is uniformly continuous if limit exists