[Apostol's Calculus] Exercises 1.5: Solution to 11

Let S be the set of all functions ƒ such that for all x in their domain, the function is increasing. In other words,
S = { ƒ | ƒ' > 0 ∀ƒ }

Assume that S is a real linear space. Then S will be closed under scalar multiplication. So, for all functions ƒ ∈ S, and all real numbers a, aƒ ∈ S also. We will show that there are some real numbers a for which aƒ ∉ S, even though ƒ ∈ S.

Proof:

Assume ƒ ∈ S. Let a ∈ R such that a < 0. Then (aƒ)' = a(ƒ)' = aƒ'. Then aƒ' will be less than zero. Therefore, aƒ ∉ S (∀ƒ ∈ S). So, S cannot possibly be a real linear space.

QED