Let X be a topological space and $f_1, . . . , f_n : X → \mathbb{R}$ be continuous functions.

Question

Let X be a topological space and $f_1, . . . , f_n : X → \mathbb{R}$ be continuous functions. Show that the sum $f_1 + . . . + f_n : X → \mathbb{R}$ is continuous.

I am not sure how to think of the sum of continuous functions from topological spaces.

Answer 1

Hint: show the result for $n=2$ (i.e. the function $f_{1}+f_{2})$, and then apply induction.

Answer 2

For real-valued functions $f_1,f_2 : X \to \Bbb{R}$, their $ \textbf{pointwise sum}$ $f_1+f_2 : X \to \Bbb{R}$ defined as $(f_1+f_2)(x) = f_1(x)+f_2(x)$. Note that $f_1 + f_2 : X \to \Bbb{R}$ can be expressed as composition of maps $X \longrightarrow \Bbb{R}\times \Bbb{R} \longrightarrow \Bbb{R}$ defined by $$ x \mapsto \Big(f_1(x),f_2(x)\Big) \mapsto f_1(x) + f_2(x). $$ Since both map are continous functions (the first map $X \to \Bbb{R} \times \Bbb{R}$ is continous because each factor is continous and the addition map $\Bbb{R}\times \Bbb{R} \to \Bbb{R}$ is continous by usual $\delta-\epsilon$ argument), then $f_1 + f_2$ is continous.