In Evans, PDE edition 2 on p68 we have a Theorem that tells us some properties about the solution to the wave equation for $n=1$. It reads:
Assume $g \in C^2(\mathbb{R})$, $h\in C^{1}(\mathbb{R})$, and define u by
$$ u(x,t) = \frac{1}{2}\left( g(x+ct)+g(x-ct) \right) + \frac{1}{2c} \int_{x-ct}^{x+ct} h(y)dy$$
with $c>0$ to solve the one dimentional wave equation given by
$$ f(x) = \left\{ \begin{array}{lr} u_{tt}-c^2 u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty)\\ u(x,0)=g(x) & \text{on } \mathbb{R} \times \{t=0 \} \\ u_{t}(x,0) = h(x) & \text{on } \mathbb{R} \times \{t=0\} \end{array} \right. . $$ Then
(1) $u \in C^2( \mathbb{R} \times [0,\infty)) $
(2) $u_{tt} - c^2 u_{xx} = 0$ in $\mathbb{R} \times (0,\infty)$
(3) $\lim_{(x,t) \rightarrow (x_0 , 0^{+})}u(x,t) = g(x_0)$ and $\lim_{(x,t) \rightarrow (x_0 , 0^{+})}u_{t}(x,t) = h(x_0)$
This proof was left to say "done by direct calculation", and I did that for (1) and (2), but with the style that Evans has been proving the (3)'s in previous similiar theorems for the heat equation and laplace's equation, I'm unsure of what to do here after fiddling around with it for a while. Can someone help? (1) and (2) are EXTREMELY direct, but I'm unsure of how to prove (3) with use of a $\delta$-$\epsilon$ proof.
Thanks.
Following Willie Wong's request:
It is not necessary to use the $\delta-\varepsilon$ formalism. It suffices to exploit the represenation formula of $u$, the continuity of $g$ and that the integral containing $h$ is continuous to infer the first identity in $(3)$. Differentiating $u$ w.r.t. to $t$ and using again the properties of $g$ and $h$ shows the second identity in $(3)$.