The graph of the even function y=f(x) consists of 4 line segments, as shown above. Which of the following statements about f is false?

(A) $\displaystyle \lim_{x \to 0}(f(x) - f(0)) = 0. $

(B) $\displaystyle \lim_{x \to 0}\frac{f(x) - f(0)}{x}=0.$

(C) $\displaystyle \lim_{x \to 0}\frac{f(x) - f(-x)}{2x}=0.$

(D) $\displaystyle \lim_{x \to 2}\frac{f(x) - f(2)}{x - 2}=1.$

(E) $\displaystyle \lim_{x \to 3}\frac{f(x) - f(3)}{x - 3}$ does not exist.