$$ S=(0,1) \cup(1,2) \cup(3,4) \text { and } T=\{0,1,2,3\} \text {. } $$

Let domain and co-domain of a function $y=f(x)$ are $S$ and $T$ respectively.

(A) There are infinitely many elements in domain and four elements in co-domain.

$\Rightarrow$ There are infinitely many functions from $S$ to $T$.

$\Rightarrow$ Option $(A)$ is correct

(B) If number of elements in domain is greater than number of elements in co-domain, then number of strictly increasing function is zero.

$\Rightarrow$ Option (B) is incorrect

(C) Maximum number of continuous functions = $4 \times 4 \times 4=64$

(Every subset $(0,1),(1,2),(3,4)$ has four choices)

$\because 64 < 120 \Rightarrow$ option (C) is correct.

(D) For every point at which $f(x)$ is continuous, $f(x)=0$

$\Rightarrow$ Every continuous function from $S$ to $T$ is differentiable.

Option (D) is correct.