Reading a book about Ramsey theory this is the first example of a compact (semitopological) semigroup, which is a topological Hausdorff space $S$ with a binary operation $*$ for which $x \mapsto x*s$ is a continuous map for all $s$ in $S$

If $X$ is a compact Hausdorff space, then the Tychonov cube $X^X$ is a compact (semitopological) semigroup with the composition from the left operation, $f \mapsto f \circ g$. Note that in general the operation of composition from the right $f \mapsto g \circ f$ is not necessarily continuous on $X^{X}$ unless $g: X \rightarrow X$ is continuous.

I've been trying to prove that the composition from the left is continuous but i still cant, a hint would help me a lot

Reading a book about Ramsey theory this is the first example of a compact (semitopological) semigroup, which is a nonempty semigroup S with compact Hausdorff topology for which $x \mapsto x*s$ is a continuous map for all $s$ in $S$

If $X$ is a compact Hausdorff space, then the Tychonov cube $X^X$ is a compact (semitopological) semigroup with the composition from the left operation, $f \mapsto f \circ g$. Note that in general the operation of composition from the right $f \mapsto g \circ f$ is not necessarily continuous on $X^{X}$ unless $g: X \rightarrow X$ is continuous.

I've been trying to prove that the composition from the left is continuous but i still cant, a hint would help me a lot