Question

let f:[0,1] - [0,1] be continuous function .show that there exist a in [0,1] such that f(a)=a

Answer 1

At some b,c in [0,1]

f(b) = 0
f(c) = 1

That is, the graph goes from the top to the bottom of the box bounded by the axes and x=1 and y=1.

At some point (or multiple points) x=a, it must cross the diagonal from (0,0) to (1,1) where f(a) = a.