Let f and g be continuous functions on 01 and suppose that f

Let f and g be continuous functions on [0,1], and suppose that f(0) g(1). Show that there is some c epsilon (0,1) such that f(c) = g(c).

Solution

Define a function:

h=f-g on the interval [0,1]

f and g are continuous on this interval so h is also continuous on [0,1]

h(0)=f(0)-g(0)<0

h(1)=f(1)-g(1)>0

So by Intermediate value theorem h must take the value 0 in the interval: (0,1)

ie h(c)=0 for some c in (0,1)

or f(c)-g(c)=0 for some c in (0,1)

Hence proved.