d'alembert's formula

Question

I'm studying the Cauchy problem for the wave equation $n=2$; $$\begin{cases}u_{tt}=\alpha^{2} u_{xx}, x \in\mathbb{R}, t>0\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} \end{cases}$$

By d´Alembert's formula we know that

$u(x,t)=\frac{f(x+\alpha t)+f(x-\alpha t)}{2} +\frac{1}{2\alpha}\int^{x+\alpha t}_{x-\alpha t}g(s)ds$

is only solution of the above problem. But in the proof, the uniqueness of $u$, second my book reference, is given by the uniqueness of the functions $f(x)\in C^{2}(\mathbb{R})$ and $g(x)\in C^{1}(\mathbb{R}).$

I do not understand this statement! Can anyone help me? thank you very much.

Answer 1

Note that $f(x)\in C^{2}(\mathbb{R})$ and $g(x)\in C^{1}(\mathbb{R})$ imply $u(x,t)\in C^{2}$. The solution at point $(x, t)$ is uniquely determined by values of $f$ and $g$ ​in the interval $[x-\alpha t,x+\alpha t]$. This interval is the Domain of Dependency of solution at point $(x,t)$.