Is there a function with such properties?

Question

Is there a monotonically increasing (decreasing), noncontinuous function with the domain $\mathbb{R}$? Or for every function, if it is monotonically increasing (decreasing), and the domain is $\mathbb{R}$, then is the function continuous on the domain (in this case $\mathbb{R}$)? The last claim seems true for me, I tried to prove it, but I have no success in it. I tried to prove this by contradiction. But I'm not asking for proof, I'm just asking: Does this seem true to you or can you find a counterexample?

Answer 1

There is a celebrated result in mathematics due to Lebesgue and it is often referred to as the Differentiation Theorem. It states:

If $f$ is monotone on some open interval $I$ then $f$ is differentiable almost everywhere on the interval $I$ (this includes also the situation when $I=\mathbb{R}$).

Almost everywhere means that a property holds for all values in the interval except for a set of values in the interval of measure zero. Consequently almost everywhere differentiability implies at least almost everywhere continuous (however not necessarily continuous everywhere). As an archetype example you can take the step function $f(x):=[x]$ where $[x]$ is the integer part of $x$. This function is monotone increasing but not continuous everywhere.