Does there exist a continuous and differentiable EVEN function whose slope at zero isn't zero?

Question

as I understand, there should not be a case, where the slope at x=0 is nonzero, if the function is even and continuously differentiable at all points including x=0.

Answer 1

Well, just compute the derivative: If $f$ is even and differentiable at $0$, we have

$$ 2f'(0)=\lim_{x\to 0} \left(\frac{f(x)-f(0)}{x}+\frac{f(-x)-f(0)}{-x}\right)=\lim_{x\to 0} \frac{f(x)-f(-x)}{x}=0, $$ since $f$ is even.