If $f$ is monotone prove it is continuous
126878
Monotonic function satisfying darboux property $\Rightarrow$ continuous
39235
Argue that the iterated integral of a continuous function is continuous
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
3772
About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$
68048
Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
12523
Supremum of a Continuous Function is Continuous
121957
Is $p\mapsto \|f\|_p$ continuous?
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
143174
property of continuous function
145368
How to show $f:Y\to X^{+}$ is continuous?
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 $\forall\alpha\in\mathbb{R}$
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
62048
Proving a Lipschitz function is continuous
112409
If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing
158888
Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
158423
How to prove that if $f$ is continuous a.e., then it is measurable.