Embedding of Sobolev space $W_2^1[a,b]$ in $C[a,b]$

Question

Let's define $$||f||_{1,2}=[\int_a^b(f(x)^2+f'(x)^2)dx]^{\frac{1}{2}}$$

and the Sobolew space $W_2^1[a,b]$ to be the completion of $C^1[a,b]$ with respect to $||f||_{1,2}$ norm. How can we show that:

$$W_2^1[a,b]\subset C[a,b]$$

Answer 1

Hint: Take the fundamental theorem of calculus and obtain the bound $$|f(x)| \le C \, \|f\|_{1,2}$$ for all $x \in [a,b]$, $f \in C^1[a,b]$ (and $C$ independent of $f$).