Proving the Existence of a Solution for f(x)=c in a Continuous Function in [a,b]

[sankalpmittal]

Homework Statement

If f(x) be a "continuous" function in interval [a,b] such that f(a)=b and f(b)=a, then prove that there exists at least one value "c" in interval (a,b) such that f(c)=c.

Note: [a,b] denotes closed interval from a to b that is a and b inclusive. (a,b) denotes open interval from a to b that is excluding a and b.

Homework Equations

Concept of continuity.

The Attempt at a Solution

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As function f(x) is continuous in [a,b] so graph of f(x) between x=a and x=b will be without any "break" and it covers value f(x) from a to b as well. Now as c lies between a and b i.e. a<c<b and f(b)=a and f(a)=b so there should be at least one solution of the equation f(x)=c. But how can we say that solution of equation f(x)=c is x=c ? How can I prove it ?

Please help !

Thanks in advanced... :)

BTW, coming back after a long time!

 

[gopher_p]

Consider the function $g(x)=f(x)-x$ on the interval $[a,b]$.

 

[sankalpmittal]

Why this function ? Question does not say it. Ummmm...

(Thanks for quick reply)

 

[gopher_p]

It's just a suggestion/hint. Note that $f(c)=c$ iff $g(c)=0$.

 

[sankalpmittal]

Please help me if i misunderstood. Question is asking us about f(x). What has g(x) to do with it and how will it answer the OP. And why we took g(x)=f(x)-x ? I do notice what you're saying though.

 

[gopher_p]

If I were to say much more, it would become less of a hint and more of me telling you how to do the problem.

Also, the suggestion was that you consider the function $g$. I made you aware of it and implied that it was maybe pertinent to answering the problem. Now your job is to sit down and think about it for a bit. Maybe write down all of the facts that you can deduce about $g$ given what you know about $f$ and the relationship between $a$ and $b$.

 

[sankalpmittal]

I do know that between x=a to x=b the graph y=x intersect the curve f(x) at at least one point. At that coordinate is (x,x). But what i am not getting is that how is x=c necessarily at least at one point ?

I have roughly sketched the figure.

 

[GFauxPas]

Are you allowed to call on the intermediate value theorem?

 

[Ray Vickson]

I do know that between x=a to x=b the graph y=x intersect the curve f(x) at at least one point. At that coordinate is (x,x). But what i am not getting is that how is x=c necessarily at least at one point ?

I have roughly sketched the figure.

You say "

I do know that between x=a to x=b the graph y=x intersect the curve f(x) at at least one point. At that coordinate is (x,x). But what i am not getting is that how is x=c necessarily at least at one point ?

I have roughly sketched the figure.

You say "I do know that between x=a to x=b the graph y=x intersect the curve f(x) at at least one point." How do you know that? That is exactly what you are trying to prove!