If f is continuous on [0,2] with f(0) = f(2), show that there is a c element of [0,2] such that f(c) = f(c+1)?

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Answer 1 · 2015-09-09T10:02:24

We consider the following function #g(x)=f(x+1)-f(x)#

Hence we have that

#g(1)=f(2)-f(1)=f(0)-f(1)# and

#g(0)=f(1)-f(0)#

If we take the product we have that

#g(0)*g(1)=-(f(0)-f(1))^2<0#

From Intermediate value theorem that means there is a c for which #g(c)=0=>f(c)=f(c+1)# qed