Prove that if $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are continuous at $x=a$ then so are the functions $\min\left \{ f,g \right \}$ and $\max\left \{ f,g \right \}$.
I know that the definition of continuity is as follows. Suppose $X$ and $Y$ are metric spaces, $E \subseteq X$, $p \in E$ and $f:E\rightarrow Y$. We say that $f$ is continuous at point $p$ if for all $\epsilon >0$, $\exists \delta >0$ such that
$$d\left ( f(x), f(p)\right ) < \epsilon$$ for all $x \in E$ for which $d\left ( x,p \right )< \delta$.
Now to show that $\min\left \{ f,g \right \}$ is continuous at$x=a$, do I have to show something like what's below?
Want to show that for all $\epsilon >0$ there exists a $\delta>0$ such that
$$d\left ( \min\left \{ f(x),g(x)\right \}, \min\left \{ f(a),g(a)\right \} \right )< \epsilon$$ for all $x$ for which $d\left ( x,a \right )< \delta$.
The converse applies for $\max\left \{ f,g \right \}$.
I'm not sure how to get started on either of the two.
Best Answer
The standard proof is to observe that $$ \max\{f,g\}=\frac{f+g}{2}+\frac{|f-g|}{2}$$ and $$ \min\{f,g\}=\frac{f+g}{2}-\frac{|f-g|}{2}$$ and then use the fact that sums and compositions of continuous functions are continuous.