An Example of a Non-continuous Injection on a Real Interval, that is Not Strictly Monotonic.

Question

Theorem

Let $I$ be a real interval. Let $f:I \to \mathbb R$ be an injective continuous real function.
Then $f$ is strictly monotone.

If the condition on continuity indeed is necessary (it most probably is but I would prefer to see it more clearly...), then we would be able to find such a function. Otherwise, I wouldn't be so convinced. I've seen the proof that uses the Intermediate Value Theorem and I do realize it requires continuity, obviously. But I don't know if we could do without it.

Answer 1

Consider $I = [0,2]$ and $f$ defined on $I$ via $f(x) := x$ when $x \leq 1$ and $f(x):= -x $ when $x > 1$.