Let $f: \mathbb R \to \mathbb R$ be a strictly monotone functions. Consider the following metric on $\mathbb R$: $$ d(x,y) = |f(x) - f(y)|, \qquad x,y \in \mathbb R. $$ Whether that metric topology is the same as the standard one?
It is the same if $f$ is continuous or has finite number of points of discontinuity. But whether it's the same for arbitrary $f$, e.g. if it discontinuous on $\mathbb Q$?
Actually, I wanted to ask, whether that metric topology always contain the standard one? It is true if there are finite number of points of discontinuity: then we have the standard topology generated by the open intervals plus we have points and semiopen (in usual topology) intervals that are open.