An increasing function $f:B\to\mathbb{R},\ B\subset\mathbb{R}$ must be continuous at every element of $B$ except for a countable subset of $B$
28191
How to show that a set of discontinuous points of an increasing function is at most countable
60700
Continuity of a function to the integers
147546
$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
62723
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
130019
Function that is continuous only on the transcendental numbers
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
53627
Proving continuity of $f$
136011
Must $f$ be continuous?
75735
Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
140929
Increasing, continuous function implies connectivity and viceversa.
9320
Extension of continuous function on a closed subset of $\mathbb{R}$
62817
Functions which are Continuous, but not Bicontinuous
35133
Every increasing function from a certain set to itself has at least one fixed point
58816
Existence of a continuous function which does not achieve a maximum.
143174
property of continuous function