Problem Statement:

Let $S=(0,1)\cup (1,2)\cup (3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?

(A) There are infinitely many functions from $S$ to $T$.

(B) There are infinitely many strictly increasing functions from $S$ to $T$.

(C) The number of continuous functions from $S$ to $T$ is at most 120.

(D) Every continuous function from $S$ to $T$ is differentiable.

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Solution:

Understanding the Sets:

• $S$ is the union of three open intervals: $(0,1)$, $(1,2)$, and $(3,4)$.

• $T$ is a finite set with four elements: $\{0,1,2,3\}$.

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Option (A): There are infinitely many functions from $S$ to $T$.

• A function $f:S\to T$ assigns to each $x\in S$ a value in $T$.

• Since $S$ is an infinite set (uncountable) and $T$ is finite, the number of possible functions is $4^{\mathrm{\mid }S\mathrm{\mid }}=4^{{\mathrm{\aleph }}_0}=\mathfrak{c}$ (the cardinality of the continuum), which is uncountably infinite.

• Conclusion: True.

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Option (B): There are infinitely many strictly increasing functions from $S$ to $T$.

• A strictly increasing function $f:S\to T$ must satisfy $x<y⟹f(x)<f(y)$.

• Since $T$ has only four elements, the range of $f$ can have at most four distinct values.

• The domain $S$ consists of three disjoint intervals. On each interval, $f$ can be strictly increasing, but the function must "jump" at the gaps (e.g., between $(0,1)$ and $(1,2)$, and between $(1,2)$ and $(3,4)$).

• The number of strictly increasing functions is limited by the number of ways to assign values from $T$ to the intervals while preserving the order. This is finite because $T$ is finite.

• Conclusion: False. There are only finitely many strictly increasing functions.

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Option (C): The number of continuous functions from $S$ to $T$ is at most 120.

• A continuous function $f:S\to T$ must be constant on each connected component of $S$ because $T$ is discrete (no nontrivial connected subsets).

• $S$ has three connected components: $(0,1)$, $(1,2)$, and $(3,4)$.

• For each component, $f$ can take any of the four values in $T$ independently.

• Thus, the total number of continuous functions is $4\times 4\times 4=64$.

• Since $64\le 120$, the statement is true.

• Conclusion: True.

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Option (D): Every continuous function from $S$ to $T$ is differentiable.

• As established, continuous functions $f:S\to T$ are constant on each connected component of $S$.

• Constant functions are differentiable everywhere on their domain.

• However, the question is phrased ambiguously. If "differentiable" is interpreted as "differentiable everywhere on $S$", then this is true because constant functions are differentiable.

• But if "differentiable" includes differentiability at the boundary points (which are not in $S$), then it's vacuously true since $S$ is open.

• Conclusion: True (under standard interpretations).

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Final Answer:

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

$$\boxed{{\displaystyle A,C,D}}$$