Prove the set of functions $f: \mathbb R \rightarrow \mathbb R$ having the following property ($\epsilon, \delta,x_1,x_2 \in \mathbb R$)
$\forall \epsilon >0 \qquad, \exists \delta>0 \qquad, (x_1-x_2) < \delta \Rightarrow |f(x_1)-f(x_2)|<\epsilon$
is the set of constant functions.
I'm failing to understand why this is true.
If $f$ is constant I can see that:
There is always a positive $\delta$ such that $x_1-x_2<\delta$ and $|f(x_1)-f(x_2)|=0<\epsilon$, but I don't see the implication. Also I can't see why this is not valid for non-continuous functions.