Is $C[a,b]$ dense in $C^1[a,b]$?

Question

is it true that the space $C[a,b]$ is dense in $C^1[a,b]$? Why or why not?

And in the other direction. Is it true that the space $C^1[a,b]$ is dense in $C[a,b]$?

Answer 1

Actually, the inverse is true.

$$ C^1[a,b]\subset C[a,b], $$ and $C^1[a,b]$ is dense in $C[a,b]$. In fact, even the set of polynomials $\mathbb P$ (as a consequence of the theorem of Stone and Weierstrass) in $[a,b]$, which is a subset $C^1[a,b]$, is dense in $C[a,b]$.

Answer 2

$C[a,b]$ is not a subset of $C^{1}[a,b]$ so we can not talk about $C[a,b]$ being dense in $C^{1}[a,b]$. The other way is true and it follows by Wierstrass approximation.