Qn : Does a continuous function always have a fixed point in [0, 1]?

[ice_kid]

I hope someone can help me wif this qnestion.

Qn : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is
x₀ ∈ [0, 1] such that f (g(x₀)) = g(f (x₀)).

Thanks in advance.

 

[awkward]

I hope someone can help me wif this qnestion.

Qn : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is
x₀ ∈ [0, 1] such that f (g(x₀)) = g(f (x₀)).

Thanks in advance.

Since both functions are onto [0,1], there are points a and b in [0,1] such that f(g(a)) = 0 and f(g(b)) = 1.

Then we must have
$$f(g(a)) - g(f(a)) \leq 0$$
and
$$f(g(b)) - g(f(b)) \geq 0$$.

The function h defined by $$h(x) = f(g(x)) - g(f(x))$$ is continuous. So...

Maybe you can take it from here.