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.
Let f(x) and g(x) be functions defined on an open interval I and let h(x) = max{f(x); g(x)} and j(x) = min{f(x); g(x)}. Suppose that h(x) and
j(x) are continuous on I. Either show that f(x) and g(x) are also continuous on I or else give a counterexample.
Any ideas on how to structure this proof using contradiction?
Hmm. Alright, so I guess the easiest way to say it then would be that \(\displaystyle \max\{f,g\}=\frac{f+g+|f-g|}{2}\), can you show that's continuous by using some of your known theorems?
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?
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?
Oh, wow! I'm sorry, I completely misread the question. Please forgive me, I have to go. I hope another member can help you. Once again, I'm sorry (Worried)
Let f(x) and g(x) be functions defined on an open interval I and let h(x) = max{f(x); g(x)} and j(x) = min{f(x); g(x)}. Suppose that h(x) and
j(x) are continuous on I. Either show that f(x) and g(x) are also continuous on I or else give a counterexample.
Any ideas on how to structure this proof using contradiction?
A better idea would be to look for a counterexample, perhaps by taking f to be a function that only takes the values 0 and 1, with a jump discontinuity at some point. Then think about how to construct g so as to make h and j continuous.
Let f(x) and g(x) be functions defined on an open interval I and let h(x) = max{f(x); g(x)} and j(x) = min{f(x); g(x)}. Suppose that h(x) and
j(x) are continuous on I. Either show that f(x) and g(x) are also continuous on I or else give a counterexample.
Any ideas on how to structure this proof using contradiction?