We have #f,g:[-1,1]->RR# two continous functions. How to demonstrate that if exist #a,b in[-1,1],a<b# such that #f(a)=g(b)# and #f(b)=g(a)# then exist #uin[-1,1]# such that #f(u)=g(u)#?

1 Answer
George C.
May 27, 2017

Let #h(x) = f(x)-g(x)# and use the Intermediate Value Theorem.

Explanation:

Let:

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

Note that #h(x)# is continuous on #[-1, 1]#.

Then:

#h(a) = f(a)-g(a) = g(b)-f(b) = -h(b)#

If #h(a) = 0# then #f(a)=g(a)# so we can put #u=a#

Otherwise, #h(a)# and #h(b)# are of opposite signs, so by Bozano's Theorem or the Intermediate Value Theorem, there is some #u# in #(a, b)# with #h(u) = 0#.

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