If $f$ is increasing on $[a,b] $and $f$ has the intermediate value property then $f$ is continuous at $x$ for every $x \in [a,b]$
77944
Injective functions with intermediate-value property are continuous. Better proof?
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
39406
Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?
57023
If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
141618
Is every real valued function on an interval a sum of two functions with Intermediate Value Property?
40915
Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
89568
Uniform continuity on $[a,b]$ and $ [b,c]$ $\implies$ uniform continuity on $[a,c]$.
121957
Is $p\mapsto \|f\|_p$ continuous?
106060
If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
65033
Prove inverse of strictly monotone increasing function is continuous over the range of original function
22061
$|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
128911
Continuous extension of a real function
112409
If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing
90619
For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
82280
Problem about continuous functions and the intermediate value theorem
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
44269
If $f$ is nonnegative and continuous on $[a,b]$, then $\left(\int_a^b f(x)^n \ dx\right)^{1/n}\to\max\limits_{[a,b]} f$