The domain $S$ is a union of three open intervals, each of which contains infinitely many real numbers. The codomain $T$ has $4$ elements.

(A) Since $S$ is uncountable and each point can independently map to any of the $4$ targets, the family of all functions from $S$ to $T$ is uncountable. Hence there are infinitely many functions. True.

(B) A strictly increasing function from $S$ to $T$ would have to take values in a totally ordered $4$-element set, yet $S$ is uncountable. A strictly increasing map on an uncountable totally ordered set cannot land in a finite set, so no strictly increasing function exists, let alone infinitely many. False.

(C) A continuous map from a connected interval into the discrete finite set $T$ must be constant on that interval. With $3$ intervals and $4$ possible constant values on each, the total count of continuous functions is $4\times 4\times 4=64\le 120$. True.

(D) Every continuous function from $S$ to $T$ is piecewise constant on each component interval. A constant function is differentiable everywhere on the open interval, hence the whole map is differentiable on $S$. True.

$\therefore$ The correct options are (A), (C) and (D).