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

Q: Wait, why can't we just pick different values for different points in the interval $(0, 1)$? Why must it be constant?

Because of the Intermediate Value Theorem! If a function is continuous on an interval, it must map that interval to a connected set. Since $T$ is a set of discrete points, the only way to satisfy continuity is if the function does not change its value at all. If it took two different values, it would have to take every value in between, which is impossible in $T$.

Q: Is a constant function really differentiable? I thought differentiability required a changing slope.

A constant function has a slope of $0$. A slope of $0$ is a perfectly valid real number. Therefore, the derivative exists and is $0$ everywhere. Differentiability does not require the function to change; it only requires the limit of the difference quotient to exist.

Q: Why did we reject the 'strictly increasing' option so quickly?

Because a strictly increasing function requires a one-to-one mapping. With infinite points in the domain and only 4 in the codomain, you would violate the Pigeonhole Principle immediately. You simply run out of values in $T$ to assign to the infinite points in $S$ while maintaining the strictly increasing condition.

Question

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 ?

Options:

(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

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a domain $S = (0, 1) \cup (1, 2) \cup (3, 4)$, which consists of three disjoint open intervals. The codomain is a discrete set $T = \{0, 1, 2, 3\}$.

The tension in this problem arises from mapping an uncountable set of points in $S$ to a finite, discrete set of points in $T$.

### Evaluating Option A: Total Number of Functions

A function $f: S 	o T$ assigns one of the 4 values from $T$ to every point in $S$. Since $S$ contains infinitely many points, we are making an infinite number of independent choices.

The total number of such functions is given by:

$$|T|^{|S|} = 4^{\infty}$$

This value is clearly infinite. Therefore, **Option A is true**.

### Evaluating Option B: Strictly Increasing Functions

For a function to be strictly increasing, the condition $x_1 < x_2$ must imply $f(x_1) < f(x_2)$.

If we attempt to map the infinite points of any interval in $S$ to the set $\{0, 1, 2, 3\}$, we would require a unique, strictly larger value for every point. Since $T$ only contains 4 elements, we would exhaust the available values immediately.

Thus, it is impossible to construct a strictly increasing function. **Option B is false**.

### Evaluating Option C: Continuous Functions

According to the **Intermediate Value Theorem**, a continuous function must map a connected interval to a connected set. Since $T$ is a discrete set, the only connected subsets of $T$ are singletons (sets containing exactly one point).

Consequently, any continuous function $f$ must be constant on each connected component of $S$. The components are $(0, 1)$, $(1, 2)$, and $(3, 4)$.

For each of the 3 components, we have 4 possible constant values to choose from. The total number of continuous functions is:

$$4 	imes 4 	imes 4 = 64$$

Since $64 \le 120$, **Option C is true**.

### Evaluating Option D: Differentiability

We have established that any continuous function $f: S 	o T$ must be piecewise constant. Specifically, on each interval, $f(x) = c$ for some $c \in \{0, 1, 2, 3\}$.

The derivative of a constant function is zero:

$$f'(x) = \frac{d}{dx}(c) = 0$$

Since the derivative exists and is equal to $0$ for all $x \in S$, the function is differentiable. Therefore, **Option D is true**.

Animated Solution for Mathematics - Limits, Continuity and Differentiability: 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 ?

Select Answer:

Visualized Solution

Visualizing Domain and Codomain

Total Number of Functions (Setup)

Total Number of Functions (Compute)

Strictly Increasing Functions (Logic)

Strictly Increasing Functions (Compute)

Continuous Functions (Logic)

Constant on Connected Components

Counting Continuous Functions

Continuous Functions Conclusion

Differentiability (Logic)

Differentiability Conclusion

The Sigma Insight: Relationship Between Continuity and Differentiability

Analyzing the Setup

Evaluating Option A

Evaluating Option B

Evaluating Option C

Evaluating Option D

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