Right continuity implies continuity on some interval

Question

I just came up with this statement and can't verify if it's true. Suppose we have a function $f:\mathbb{R}^{\geq0} \rightarrow \mathbb{R}$. I know that right continuity means $\lim_{\epsilon \downarrow 0} f(x+\epsilon)=f(x)$ and continuity - $\lim_{\epsilon \rightarrow 0} f(x+\epsilon)=f(x)$ and I'm also familiar with the $\epsilon - \delta$ definitions. I tried to construct a counter example by taking a step function on $[0,1]$ with partition $0<\frac{1}{n}<\frac{2}{n}...<1$ and letting $n \rightarrow \infty$. The idea was to construct a sequence tending to $0$ such that the function would have discontinuities at each element of the sequence and then do a similar procedure for any of the intervals. Any help would be appreciated

Answer 1

I'm fairly certain that any countable subset $D$ of $\mathbb R$ can be the discontinuity set for an everywhere right continuous function from $\mathbb R \rightarrow \mathbb R.$ Here is an example for $D = \mathbb Q,$ which suffices to disprove your conjecture.

Let $\{q_n\}$ be a (bijective) enumeration of $\mathbb Q.$ We define $f: \mathbb R \rightarrow \mathbb R$ as follows. For each $x \in \mathbb R,$ let $f(x)$ be $\sum 2^{-n},$ where the summation is over all values of $n$ such that $q_n \leq x.$ In [1], the discussion on pp. 32-33 and Exercise 34 on p. 33 indicate (among other things) that (i) $f$ is everywhere right continuous, and (ii) $f$ is discontinuous (i.e. not left continuous) at each $x \in \mathbb Q.$

If instead we define $g: \mathbb R \rightarrow \mathbb R$ by letting $g(x)$ be $\sum 2^{-n},$ where the summation is over all values of $n$ such that $q_n < x,$ then in [2], Exercise 8.26 on p. 113 asks the reader to show (among other things) that (i) $g$ is everywhere left continuous, and (ii) $g$ is discontinuous (i.e. not right continuous) at each $x \in \mathbb Q.$

Incidentally, both $f$ and $g$ have the additional property of being strictly increasing on $\mathbb R.$

[1] Neal L. Carothers, Real Analysis, Cambridge University Press, 2000.

[2] Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis, Springer-Verlag, 1969.