Does such function exist?

[Grigorios Kostakos]

Does exist a real function \(f:[a,b]\longrightarrow{\mathbb{R}}\) which is bounded, monotonic and discontinuous in uncountable many points of \([a,b]\) ?

[Demetres]

No there isn't even if $f$ is allowed to be unbounded. So let $f$ be a monotonic function. We will show that it has at most countably many discontinuities. Let us also suppose without loss of generality that $f$ is monotone increasing.

Since $f$ is monotonic, then it has left and right limits at each $x \in (a,b)$. Let $x_{\ell},x_r$ be the corresponding limits. If $f$ is discontinuous at $x$, then $x_{\ell} < x_r$. Furthermore, since $f$ is increasing, then it cannot take any value in the interval $(x_{\ell},x_r)$. Let $q_x \in (x_{\ell},x_r) \cap \mathbb{Q}$.

So we have exhibited for each point of discontinuity $x$ of $f$ (except perhaps at $x=a$ and $x=b$) a rational number $q_x$. Furthermore we have $x < y \iff q_x < q_y$.

Thus $f$ can only have countably many discontinuities.