Provide an example where $S$ is not path connected, $f:S\longrightarrow T$ is continuous and Im$f$ is path connected.

Question

$\DeclareMathOperator{\Im}{Im}$

Provide an example where $S$ is not path connected, $f:S\longrightarrow T$ is continuous and $\Im{f}$ is path connected.

Let $S = [0,1] \cup \{2\} \subset \mathbb{R}$, $T=\mathbb{R}$. Let $f:S \longrightarrow T$ be given by, $$f(s) = \begin{cases} s & 0\leq s \leq 1 \\ s-1 & s=2 \end{cases} $$

Then $f$ is continuous and $\Im{f}=[0,1]$ is obviously path connected.

Is this example correct? Are there simpler examples?

Answer 1

A simpler example: take $f$ constant.

Answer 2

Take $Y=\Bbb R$, usual topology (or any path-connected space), $X=\Bbb R$ in the discrete topology (totally disconnected as always), $f(x)=x$. Any function on a discrete space is continuous.