2. (2007 Zhejiang Province High School Mathematics Competition B Paper) Let the non-constant function \( f(x) (x \in \mathbf{R}) \) be an even function, and its graph is symmetric about the line \( x = \frac{\sqrt{2}}{2} \). Then the function is
A. A non-periodic function
B. A periodic function with a period of \( \frac{\sqrt{2}}{2} \)
C. A periodic function with a period of \( \sqrt{2} \)
D. A periodic function with a period of 2
2. $\mathrm{C}$ Because an even function is symmetric about the $y$-axis. If the graph of the function $y=f(x)$ is symmetric about the line $x=\frac{\sqrt{2}}{2}$, then $f\left(\frac{\sqrt{2}}{2}-x\right)=$ $f\left(\frac{\sqrt{2}}{2}+x\right)$, which means $f(x+\sqrt{2})=f\left[\frac{\sqrt{2}}{2}+\left(\frac{\sqrt{2}}{2}+x\right)\right]=f(-x)=f(x)$, hence the function is a periodic function with a period of $\sqrt{2}$. The correct choice is $\mathrm{C}$.