Is every nondecreasing function necessarily a pointwise limit of strictly increasing functions?

Question

Suppose I have a function $f$ defined on the compact convex set $\mathcal{D} \subseteq \mathbb{R}$, with the values in $\mathbb{R}$, and this functions is nondecreasing. Is it necessarily a pointwise limit of some sequence of strictly increasing functions?

Answer 1

Yes:$$(\forall x\in\mathcal{D}):f(x)=\lim_{n\to\infty}\left(f(x)+\frac xn\right).$$

Answer 2

The first answer is pefectly fine, but we can also archieve uniformly convergence for non-compact sets.

We can assume that $f$ is defined on $\mathbb{R}$ and is nondecreasing.

First note, that $\mathcal{D}$ have to be an interval [m,M], where $m = \inf \mathcal{D} \in \mathcal{D}$ and $M = \sup \mathcal{D} \in \mathcal{D}$, because of the compactness and convexity. We may also note that any montone-increasing function can be extened to $\mathbb{R}$ by just defining$$g(x) = \inf_{y \geq x, y\in \mathcal{D}} f(x).$$

Another example for approximate sequence, which also converges uniformly, is $$f_n(x) =f(x) + \frac{1}{n} \arctan(x).$$