Hard continuity problem/proof  TOPIC_SOLVED

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Hard continuity problem/proof

Postby smad on Mon Oct 15, 2012 10:35 pm

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.
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Re: Hard continuity problem/proof

Postby nona.m.nona on Mon Oct 15, 2012 11:42 pm

smad wrote: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?

If you would care for further guidance, kindly please reply with your answers to these questions, along with your thoughts thus far. Thank you.
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Re: Hard continuity problem/proof

Postby smad on Tue Oct 16, 2012 7:18 am

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).
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Re: Hard continuity problem/proof  TOPIC_SOLVED

Postby nona.m.nona on Wed Oct 17, 2012 12:22 am

smad wrote: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)?
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