Continuous Functions

**sssitex** · January 22nd 2011, 09:29 AM

I have difficulties solving the following exercise:

Prove that if f and g are continuous functions from a topological space X to $\mathbb{R}$ , then f+g and fg are continuous.

The hint says: Apply this theorem: let $X_0$ , $X_1$ and $X_2$ be topological spaces and let $f:X_0 \Rightarrow X_1$ and $g: X_1 \Rightarrow X_2$ be continuous functions. Then g(f): $X_0 \Rightarrow X_2$ is continuous.

and use the easy facts that the maps (x,y) to x+y and xy are continuous (where defined).
I really don't know how to use the hint, i can't imagine that this exercise is difficult.

Thanks for help.

**Tinyboss** · January 22nd 2011, 10:07 AM

Okay, they've given you that the maps $(x,y)\mapsto xy$ and $(x,y)\mapsto x+y$ are continuous maps from $\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ . A property of the product topology is that a composite function is continuous if and only if its component functions are. Use the fact that a composition of continuous functions is continuous and you're done.

Edit: That might not be clear. The point is that the map $(f+g):X\times X\to\mathbb{R}$ is a composition of two maps, $F:X\times X\to\mathbb{R}\times\mathbb{R}$ where $F(x_1,x_2)=(f(x_1),g(x_2))$ and $s:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ , where the "s" stands for "sum", and $s(x,y)=x+y$ . They've told you s is continuous, and you know F is continuous since its components are continuous and its codomain has the product topology. Therefore the composition $s(F(x,y))$ is continuous.

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