Problem
Let S = (0, 1) U (1, 2) U (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
Solution
Let’s pick each option, one by one.
Option A
This is somewhat obvious. Each interval in S has infinitely many points, each of which can be randomly mapped to an element in T, leading to infinitely many functions. So, option A is true.
Option B
This is also somewhat trivial. Since there are only four elements in T, and infinitely many elements in S, any function that we create will be many one, and therefore cannot be strictly increasing. So, option B is false.
Option C
The only way we can make a continuous function is to map each interval in S to one of the values in T, i.e. a constant function for each interval.
For example (0, 1) maps to 1, (1, 2) maps to 3, and (3, 4) maps to 0. Given this restriction, how many continuous functions exist?
That’s fairly easy to count. Each interval in S can be mapped to any of the 4 numbers in T. Since there are three different intervals, the total number of continuous functions would be 4 x 4 x 4 or 64.
Since 64 is less than 120, option C is true.
Option D
Again, somewhat trivial. Since each continuous function is constant in each interval, it must be differentiable. Note that we do not need to worry about the differentiability at the end points, i.e. x = 0, 1, 2, 3, or 4, since these points are not in the domain of the function.
Comments
I particularly found option C weird. Somehow, the statement “64 is at most 120” doesn’t sound right, even though it is mathematically true. Instead of ‘at most 120‘, maybe ‘equal to 64‘ or ‘more than 36‘ would’ve been better? Otherwise, a decent theoretical problem on functions.