# Modification on Epsilon-Delta Definition of Continuity - Seeking a Discontinuous Function

I'm looking for a function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ such that f is not continuous, and moreover $$\forall \epsilon > 0, \exists \delta > 0$$ such that $$|f(x) - f(y)| < \epsilon \implies |x - y| < \delta$$.

I can't seem to construct such a function, yet I know this should be true, given that the epsilon-delta definition of continuity has the implication the other way around. There is a similar question here but with different quantifiers.

Let $$f(x)=x$$ if $$x\leq 0$$ and $$f(x)=x+1$$ if $$x>0$$. Then $$f$$ is discontinuous at $$0$$ but satisfies your $$\epsilon$$-$$\delta$$ condition with $$\delta=\epsilon$$.
$$1)$$ Any discontinuous monotonic function
$$2)$$ A function $$f$$, $$f(x)=x$$ for $$x\in\mathbb{Q}$$, $$f(x)=x+1,x\in\mathbb{R}-\mathbb{Q}$$