The following is a question which I need some help to solve. (Indiana University Qual 1999)
Question
Let $f : [0,1]\times[0,1] \rightarrow \mathbb{R}$ be a continuous function and define $g : [0,1] \rightarrow \mathbb{R}$ by $g(x)=\max_{y\in [0,1]} f(x,y).$Show that $g$ is continuous.
End of question.
This is what I have tried. Because of continuity (detail is routine and is obmitted here), for $x_0\in [0,1]$, there is a $y_0\in [0,1]$ such that $g(x_0)=f(x_0,y_0)$, then for $x\in[0,1]$,
$|g(x)-g(x_0)|=|\max_{y\in [0,1]} f(x,y)-f(x_0,y_0)|$ $\leq |\max_{y\in [0,1]} f(x,y)-f(x, y_0)|+|f(x,y_0)-f(x_0,y_0)|$
I can made this term $|f(x,y_0)-f(x_0,y_0)|$ small, but I don't seems to have control over $|\max_{y\in [0,1]} f(x,y)-f(x, y_0)|$, any comment?