Is $H_0^1([a,b]) \subset C([a,b],\mathbb{R})$?

Question

i have a small question : how to see that $H_0^1([a,b])\subset C([a,b],\mathbb{R})$?

Please

Thank you

Answer 1

Hint: for any $[x,y] \subseteq [a,b]$

$$ f(y) - f(x) = \int_x^y f'(s) \mathrm{d}s \leq \int_x^y |f'(s)| \mathrm{d}s \leq \|f'\|_{L^2([x,y])} \|1\|_{L^2([x,y])} $$

so $f$ is in fact uniformly continuous.

Answer 2

I am not sure what you mean by $H^1_0$ being a subset of continuous functions but I am guessing what you need is the Sobolev embedding theorem (see Adams and Fournier or McLean) which essentially states the following:

Suppose $0< \mu <1$. If $u \in H^{n/2+ \mu}(R^n) $, then $u$ is almost everywhere equal to a Holder-continuous function. Embedding of $H^1_0$ in one dimension follows from this result immediately.