Sorry reposted:
Let f(x) and g(x) be fcns defned on an interval I and let h(x) = max(f(x); g(x)) and j(x) = min(f(x); g(x)). h(x) and
j(x) are continuous on I. Show f(x) and g(x) are continuous on I or give a counterexample.
Sorry reposted:
Let f(x) and g(x) be fcns defned on an interval I and let h(x) = max(f(x); g(x)) and j(x) = min(f(x); g(x)). h(x) and
j(x) are continuous on I. Show f(x) and g(x) are continuous on I or give a counterexample.
Well, we know that the function h(x) is continuous, so the max(f,g) would be continuous. I know that there are rules for algebraic combinations of continuous functions being continuous, so I'm wondering if the max function can be worked backwards to prove f and g continuous that way. Is that where you were going with this?
Or simply take these two easy functions on [0,1]
f(x)=1 if $\displaystyle 0\leq x<\frac{1}{2} $
-1 if $\displaystyle \frac{1}{2} \leq x \leq 1$
g(x)= -1 if $\displaystyle 0\leq x <\frac{1}{2}$
1 if $\displaystyle \frac{1}{2} \leq x \leq1$
both are discontinuous but max and min functions h,j are constant hence continuous