Properties of Continuous Functions Between Given Sets

To determine which of the given statements are true, let us analyze each option individually.

Analysis of Option (A):

The set $S = (0,1) \cup (1,2) \cup (3,4)$ is an uncountable infinite set, and $T = \{0,1,2,3\}$ is a finite set containing 4 elements. The number of functions from $S$ to $T$ is given by $|T|^{|S|} = 4^{\mathfrak{c}}$, where $\mathfrak{c} = 2^{\aleph_0}$ is the cardinality of the continuum. Since $4^{\mathfrak{c}} > \aleph_0$, there are infinitely many functions from $S$ to $T$.

Thus, Option (A) is TRUE.

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Analysis of Option (B):

A function $f: S \to T$ is strictly increasing if for all $x_1, x_2 \in S$, $x_1 < x_2 \implies f(x_1) < f(x_2)$.

Suppose such a function exists. Consider an infinite sequence of distinct points $x_1 < x_2 < x_3 < x_4 < x_5$ in the interval $(0,1) \subset S$. If $f$ is strictly increasing, then: $f(x_1) < f(x_2) < f(x_3) < f(x_4) < f(x_5)$ This requires $f(S)$ to contain at least 5 distinct elements. However, $T = \{0,1,2,3\}$ contains only 4 elements.

This contradiction shows that no strictly increasing function from $S$ to $T$ can exist. Therefore, the number of strictly increasing functions is $0$.

Thus, Option (B) is FALSE.

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Analysis of Option (C):

The domain $S$ consists of three disjoint, connected open intervals: $I_1 = (0,1), \quad I_2 = (1,2), \quad I_3 = (3,4)$

Since the continuous image of a connected set must be connected, and the only connected subsets of the discrete set $T = \{0,1,2,3\}$ are its singleton sets $\{0\}, \{1\}, \{2\}, \{3\}$, any continuous function $f: S \to T$ must be constant on each connected component of $S$.

Therefore, a continuous function $f: S \to T$ is uniquely determined by choosing a single value in $T$ for each interval:
$f(x) = \begin{cases} c_1 & \text{for } x \in (0,1) \\ c_2 & \text{for } x \in (1,2) \\ c_3 & \text{for } x \in (3,4) \end{cases}$
where $c_1, c_2, c_3 \in T$.

Since there are 4 choices for each of $c_1, c_2,$ and $c_3$, the total number of continuous functions from $S$ to $T$ is: $4 \times 4 \times 4 = 4^3 = 64$

Since $64 \le 120$, the statement that the number of continuous functions is at most 120 is correct.

Thus, Option (C) is TRUE.

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Analysis of Option (D):

As established in Option (C), every continuous function $f: S \to T$ is locally constant on each open interval $I_1, I_2, I_3$.

For any $x_0 \in S$, there exists an open neighborhood $U \subset S$ containing $x_0$ such that $f(x) = c$ (a constant) for all $x \in U$. Consequently, the derivative exists at every point in $S$ and is given by: $f'(x) = 0 \quad \forall x \in S$

Hence, every continuous function from $S$ to $T$ is differentiable on $S$.

Thus, Option (D) is TRUE.

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Conclusion:

The correct statements are (A), (C), and (D).