Uniformly convergent sequence $(g_n)_{n \in \mathbb{N}}$ with each $g_n$ having local min at $0$. Does $g$ have local min at $0$?

Question

Say I have a sequence of real-valued functions on $\mathbb{R}$ $(g_n)_{n \in \mathbb{N}}$ with $, g_n \xrightarrow[]{\text{uniform}} g$. Say each $g_n$ has a local minimum at $x = 0.$ Will $g$ have a local minimum at $x = 0$ too?

I suspect the answer is no (because not continuity requirements are imposed), but I can't construct a good counterexample. Help?

Answer 1

You don't need complicated piecewise definitions, and you can have continuous (even smooth) functions: $$g_n(x)=-x^4+\frac1n\frac{x^2}{x^2+1}$$ has a local minimum at $0,$ while $g(x)=-x^4$ has a global maximum, there.