$f$ increasing , $c \in I$ ; $a_n<c<b_n , \forall n\in \mathbb N$ and $\lim \Big(f(a_n)-f(b_n) \Big)=0 $ , then $f$ is continuous at $c$ ?
53738
Monotone functions and continuity
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
29180
If $f(x/n)\to0$ when $n\to\infty$, for every $x$ and $f$ is continuous, then $f(x)\to0$ when $x\to0$, or not?
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
53627
Proving continuity of $f$
120406
$f$ strictly increasing does not imply $f'>0$
50103
Show that a continuous function satisfying $f(2^x)=f(3^x)$ for $x\in (0,\infty)$ is a constant function
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
10747
Given function $g$ continuous in line $[0,\infty)$...
153854
$X_{n}$ converges in probability, $f$ is continuous, $f(X_{n})$ converges in probability
121957
Is $p\mapsto \|f\|_p$ continuous?
129551
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
151410
Inequality of continuous functions
108060
Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞
113975
Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function
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 $\mathbb{R}$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point