Solution
= Solution
The constant maps form a connected subspace homeomorphic to $S$. Every $f\in C([0,1],S)$ is joined to the constant map $f(0)$ by
$$
H(s)(t)=f((1-s)t).
$$
Uniform continuity of $f$ makes $s\mapsto H(s)$ continuous in the uniform metric. Thus the function space is a union of connected sets, each meeting the connected set of constant maps, and so
$$
\boxed{C([0,1],S)\text{ is connected}.}
$$