Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?

Question

Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?

Answer 1

Yes.

The key is that when you (as you say in the comments) get the two scenarios:

$$ f \star (D g) = D(f \star g) = (D f) \star g $$

then you get to choose which!

So if $D f$ doesn't make sense, then you can ignore it and choose to use the identity $D(f \star g) = f \star (D g)$.

Taking this to the extreme, you get the - bizarre, in my opinion - result that if $p$ is a polynomial, then $f \star p$ is always a polynomial.