# Prove that the set, C, of continuous functions is not a subspace of the differentiable functions.

I cannot for the life of me figure out this problem:

Prove that the set, C, of continuous functions on the interval (-1, 1) is not a vector subspace of the set, D, of differentiable functions on that interval.

Maybe I'm completely making this up, but I thought all differentiable functions were continuous.

How do I attack this problem?

The question is titled: Looking for a way to pass the time?

• All differentiable functions are continuous, but not all continuous functions are differentiable. There are easy examples. – Brian M. Scott Dec 2 '14 at 5:59
• You just need to find one continuous function on $(-1,1)$ that is not differentiable on $(-1,1)$. Check out the Weierstrauss function – graydad Dec 2 '14 at 6:02
• @graydad: That’s working awfully hard for an example. What about the familiar function whose graph is a $\lor$, more or less? – Brian M. Scott Dec 2 '14 at 6:02
• @BrianM.Scott No questions there that that is a much easier example. My comment was biased, as I am very intrigued by the properties of the Weierstrauss function – graydad Dec 2 '14 at 6:05
• What do you mean, "The question is titled: Looking for a way to pass the time?"? – Gerry Myerson Dec 2 '14 at 6:22

Just pick your favorite function that is continuous on $(-1,1)$, but not differentiable at some point in $(-1,1)$, like $f(x)=|x|$ or $g(x)=x^{2/3}$ (neither is differentiable at $x=0$).
The upshot here is that if $X$ is not a subset of $Y$, then $X$ cannot be a subspace of $Y$.