## Hard continuity problem/proof

Limits, differentiation, related rates, integration, trig integrals, etc.
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### Hard continuity problem/proof

Let f and g be two continuous functions on the closed interval [0,1] such that for every x in [0,1] : f(x)<g(x).

Prove that there exists some number m>0 such that for every x in [0,1] : f(x)+m<g(x).

Well i think i have to use the fact that every continuous function on a closed interval is bounded by a maximum and a minimum, but since I've never dealt with anything like this I can't really start it. Thanks for the help before hand.

nona.m.nona
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### Re: Hard continuity problem/proof

Let f and g be two continuous functions on the closed interval [0,1] such that for every x in [0,1] : f(x)<g(x).

Prove that there exists some number m>0 such that for every x in [0,1] : f(x)+m<g(x).
Since f and g are each continuous, what might be said about the continuity of h = g - f? What must be true of the sign of h on the interval? What does the Extreme Value Theorem say about a minimum value for h on the interval? How might this be useful in your proof?

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### Re: Hard continuity problem/proof

Well if f and g are both continuous than therefore h=g-f is also continuous as a difference of 2 continuous functions and h>0 because f(x)<g(x). And the extreme value theorem states that if h is a continuous function on a closed interval [a,b] then there exists a c and a d such that h(c)>=h(x)>=h(d).

nona.m.nona
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### Re: Hard continuity problem/proof

Well if f and g are both continuous than therefore h=g-f is also continuous as a difference of 2 continuous functions and h>0 because f(x)<g(x). And the extreme value theorem states that if h is a continuous function on a closed interval [a,b] then there exists a c and a d such that h(c)>=h(x)>=h(d).
What can you say about the sign of h(d)/2? What can you say about the equality (or otherwise) of h(d)/2 and h(x)? If you rename h(d)/2 as "m", what can you then conclude about f(x) and g(x)?