I try to show that for every space $X$, $(C_p(X), +)$ where $$+:C_p(X)\times C_p(X)\to C_p(X):(f,g)\mapsto f+g$$ and for every $x\in X$, $(f+g)(x)=f(x)+g(x)$ is a topological group.
The family $$\{O(f, x_1,\ldots, x_n, \epsilon) : n \in\Bbb N, x_1,\ldots, x_n \in X,\epsilon > 0\}\;,$$ Where $$O(f, x_1,\ldots, x_n,\epsilon) =\{g \in C_p(X) : \vert g(x_i)- f(x_i) \vert<\epsilon \; \text{for all }i\leq n\}$$ is a local base of $C_p(X)$ at $f$. How can we show that the inverse function is continuous?