Let `f:[0,1] rarr [0,1]` be a continuous function. Then prove that `f(x)=x` for at least one `0lt=xlt=1.`

Explanation

Let us take `g(x)= f(x) -x`
`g(0) = f(0) – 0 = f(0) >= 0 { 0<=f(x)<= 1}`
`g(1) = f(1) – 1 <= 0`

`=> g(0) * g(1)<= 0`

`=> g(x) = 0` has atleast one root in `[0, 1]`
`=> f(x) = x` for atleast one `x epsilon[0, 1]`