Using the Definition: Showing $u \in C^0 [0,1]$

[evinda]

Hello! (Wave)

I want to show the embedding $W^{1,p}(0,1) \subset C^0 [0,1]$.

So we pick a $u \in W^{1,p}(0,1)$ and want to show that $u \in C^0 [0,1]$.

Let $x_n \to x$. We want to show that $u(x_n) \to u(x)$.

Since $u \in W^{1,p}(0,1)$ we have that $u \in L_p$ and $u' \in L_p$.

And you told me to use the fact that $u(x+h)-u(x)=\int_{x}^{x+h} u'(t) dt$.

We know that a function $f$ is continuous in $x_0$ if $\forall \epsilon>0 \exists \delta>0$ such that $\forall y \in [x,x+h]$ with $|y-x_0|< \delta$ it holds that $|f(y)-f(x_0)|< \epsilon$.

Can we use this definition for $f=u, x_0=x, y=x+h$ ?

 

[Opalg]

Hello! (Wave)

I want to show the embedding $W^{1,p}(0,1) \subset C^0 [0,1]$.

So we pick a $u \in W^{1,p}(0,1)$ and want to show that $u \in C^0 [0,1]$.

Let $x_n \to x$. We want to show that $u(x_n) \to u(x)$.

Since $u \in W^{1,p}(0,1)$ we have that $u \in L_p$ and $u' \in L_p$.

And you told me to use the fact that $u(x+h)-u(x)=\int_{x}^{x+h} u'(t) dt$.

We know that a function $f$ is continuous in $x_0$ if $\forall \epsilon>0 \exists \delta>0$ such that $\forall y \in [x,x+h]$ with $|y-x_0|< \delta$ it holds that $|f(y)-f(x_0)|< \epsilon$.

Can we use this definition for $f=u, x_0=x, y=x+h$ ?

An internet search shows that this result is not easy. It seems to require an application of the Sobolev embedding theorem to show that the function $u$ satisfies a Hölder continuity condition. That in turn implies continuity in the usual sense.

Difficulties in dealing with this problem include the facts that (1) the derivative $u'$ is defined in a weak, distributional, sense (if $u$ was differentiable in the classical sense then it would certainly be continuous); and (2) functions in the space $W^{1,p}(0,1)$ are only defined up to almost everywhere equivalence, so you may have to adjust them on a null set in order to make them continuous.

 

[evinda]

An internet search shows that this result is not easy. It seems to require an application of the Sobolev embedding theorem to show that the function $u$ satisfies a Hölder continuity condition. That in turn implies continuity in the usual sense.

Difficulties in dealing with this problem include the facts that (1) the derivative $u'$ is defined in a weak, distributional, sense (if $u$ was differentiable in the classical sense then it would certainly be continuous); and (2) functions in the space $W^{1,p}(0,1)$ are only defined up to almost everywhere equivalence, so you may have to adjust them on a null set in order to make them continuous.

Couldn't we just show that $u$ is absolutely continuous?