Provide a direct proof of the following version of the Inter

Provide a direct proof of the following version of the Intermediate Value Theorem: Let f:[a,b]-->R be continuous and f(a)>f(b). Then for any L, f(b)<L<f(a) there is a c in (a,b) such that f(c)=L. Do not use the notion of connectedness or the proof for the case when f(a)<f(b).   

Solution

Given that f is continuous and f(a) >f(b).

Hence there exists L between f(a) and f(b)

Let f(b)<L<f(a)

As per our intermediate value theorem,

if f(b)<f(a) then there exists a c in (a,b) such that f(c) = L

Hence proved.