 Solution

Connected means having no separation into two nonempty disjoint open subsets. Images of intervals under $f$ are intervals. Thus for $a<b<c$, injectivity forces $f(b)$ to lie between $f(a)$ and $f(c)$, so the quoted criterion makes $f$ monotone. A monotone bijection $\mathbb{R}\to \mathbb{R}$ has no jumps, since a jump would omit an interval; hence it is continuous.