Let f g a b rightarrow c d continuous functions Suppose that

Let f, g: [a, b] rightarrow [c, d] continuous functions. Suppose that f is onto. Prove that there is a point x_0 elementof [a, b] such that f(x_0) = g(x_0).

Solution

f is onto so there is u and v in [a,b] so that

f(u)=c,f(v)=d

If, g(u)=c then proof is done

else, g(u)=c\'>c

So, f(u)-g(u)=c-c\'<0

If g(v)=d then proof is done

else, g(v)=d\'<d

f(v)-g(v)=d-d\'>0

Now consider the function

h(x)=f(x)-g(x)

So, either h(x)=0 at u or v

or h has a sign change going from u to v

And hence by Intermediate Value Theorem

h(x) =0 for some d in (u,v)

Hence proved