Proof:

Assume $X$ is path connected but not connected. Then there is a non constant continuous function $g : X \longrightarrow \mathbb{Z}$ say $f(x) = m$ and $f(y) = n$ for a pair of distinct integers $m$ and $n$. But there is also a continuous function $f : [0. 1] \longrightarrow X$ such that $g(0) = x$ and $g(1) = y$. The composite function $f\circ g : [0, 1] \longrightarrow \mathbb{Z}$ is non constant which is a contradiction.