prove that if f(x) and g(x) are continues
so are those:
http://img508.imageshack.us/img508/1639/64747667bx8.gif
??
prove that if f(x) and g(x) are continues
so are those:
http://img508.imageshack.us/img508/1639/64747667bx8.gif
??
Hint: Consider writing $\displaystyle \varphi(x)=\max\left\{f(x),g(x)\right\}$ as a piecewise function where the splits occur where the max changes from being $\displaystyle \varphi(x)=f(x)$ to $\displaystyle \varphi(x)=g(x)$.
For example consider the function $\displaystyle \varphi(x)=\max\left\{x,2x-1\right\}$ on the interval $\displaystyle [0,2]$. It is clear that $\displaystyle \varphi(x)$ may be written as $\displaystyle \varphi(x)=\left\{ \begin{array}{rcl} x & \mbox{if} & 0\leqslant x \leqslant 1\\
2x-1 & \mbox{if} & 1<x\leqslant2 \end{array} \right.$