The iteration,which is only locall convergent,does it has at least 2 fixed point?

Question

When I draw the figure of the iteration which is just locally convergent，it feels like that ,there must be at least 2 fixed points.So, I have the hypothesis

Assume $g(x)\in C^{1}(\mathbb{R})$,the iteration sequence $x_{n+1}=g(x_n)$ is locally convergent,then $g(x)$ has at least 2 fixed points.

But I have no idea how to verify it.