So I'm putting together a presentation in topology on BCT and I have the following theorems:
Let $D$ be the set of somewhere differentiable functions $[0,1] \to \mathbb R$. Then $D$ is meagre in $C[0,1]$.
Let $f \colon [0,1] \to \mathbb R$ be a differentiable function with derivative $f'$. Let $X$ be the set of points for which $f'$ is discontinuous. Then $X$ is meagre in $[0,1]$.
My temptation to rationalise these intuitively (as a taster in the introduction) as follows:
Most continuous functions are nowhere differentiable.
Derivatives are continuous at most points.
This is based off my understanding that a set $A$ is meagre in $B$ if it is small relative to $B$ in a precise sense. But there are instances where a meagre set can be quite large (even of full measure), and I've been told that it's not really correct to say "most" in this context. However, I've seen quite a lot of posts on here giving similar intuitive restatements.
I thought I'd ask here for clarification - is it proper/well-understood to say "most" here? I could say "many" but that doesn't really feel as punchy. If it's not proper to say most here, is there any better way to put it? My topological intuition is still fairly weak.