# If $f:[0,1] \rightarrow \Bbb R$ is a strictly increasing discontinuous function, can $f([0,1])$ be a subset of $\Bbb Q$?

If $$f:[0,1] \rightarrow \Bbb R$$ is a strictly increasing discontinuous function, can $$f([0,1])$$ be a subset of $$\Bbb Q$$?

Attempt: Since $$f$$ is a discontinuous strictly increasing function, this means that :

$$1.~ \forall a \in [0,1]: \lim_{x \rightarrow a^-} f(x)$$ and $$\lim_{x \rightarrow a^+} f(x)$$ exist

$$2.~$$ There can be no removable discontinuity. So, if $$f$$ is discontinuous at a point $$a :~\lim_{x \rightarrow a^-} f(x) \ne\lim_{x \rightarrow a^+} f(x)$$

$$3.~$$ There can be only a countable number of discontinuities.

Since there can only be a countable number of discontinuities, and $$[0,1]$$ is uncountable, $$f$$ must be continuous on an uncountable number of points. Thus, $$f$$ must traverse an interval by being continuous at these points and hence, cannot be a subset of rationals because each interval contains uncountable irrationals.

Is my approach correct?

• $f$ doesn't have to be discontinuous. As long as it is strictly increasing, $f([0,1])$ can't be a subset of $\Bbb Q$, by the argument in @Integrand's answer. (What is a "discontinuous function" anyway? Do you just mean that it is not everywhere continuous?) May 3, 2020 at 20:37

This is a naive thought but here goes: since $$f$$ is strictly increasing on $$[0,1]$$, if $$x, $$f(x). In other words, $$f$$ is injective. But then we couldn't have $$f([0,1])\subseteq \mathbb{Q}$$ because $$|\mathbb{Q}|<|[0,1]|$$.
• @TonyK I think a reasonable definition can be provided that $f$ is not strictly increasing at a point if it is not strictly increasing in any neighborhood of such point. May 3, 2020 at 21:13
• @TonyK: there is a definition of increasing at a point. Let $f$ be defined in some neighborhood of $c$. Then $f$ is said to be increasing at $c$ if there is a neighborhood $I$ of $c$ such that $x\in I, x<c\implies f(x) \leq f(c)$ and $x\in I, x>c\implies f(x) \geq f(c)$. The strict inequality lead to definition of strictly increasing. May 4, 2020 at 7:09