Functions and Continuity
DAILY_CHALLENGE
Grade None
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?
There are infinitely many functions from $S$ to $T$
There are infinitely many strictly increasing functions from $S$ to $T$
The number of continuous functions from $S$ to $T$ is at most 120
Every continuous function from $S$ to $T$ is differentiable
Step-by-Step Solution
Key Concept: Continuous functions from a disconnected set to a discrete set must be locally constant
(A) TRUE: $S$ is uncountable, $T$ is finite with 4 elements. There are $4^{|S|}$ = infinitely many functions.
(B) FALSE: A strictly increasing function from $S$ (which has uncountably many points in each connected component) to $T$ (discrete, finite) is impossible — within any connected interval of $S$, a strictly increasing function would need to take distinct values from $T$ for uncountably many inputs, but $T$ has only 4 elements.
(C) TRUE: A continuous function from $S$ to the discrete set $T$ must be constant on each connected component of $S$. $S$ has 3 components: $(0,1),(1,2),(3,4)$. Number of such functions $= 4^3 = 64 \leq 120$.
(D) TRUE: Any continuous function from $S$ to $T$ is constant on each component, hence has derivative $= 0$ everywhere — so it is differentiable.
Correct Answer: A, C, D