Re: How to prove the continiuity of the function.

On 1月14日, 上午8时12分, Hero Wanders <herojo...@xxxxxxxx> wrote:
Hello,

Cp is Cantor set in real analysis. m is Lebesgue measure. x ∈ [ 0, 1].

m( [0,x] ∩ ( [0,1] - Cp ) )
f (x) = -------------------------------------------------------
m( [0, 1] - Cp )

f (0) =0 , f (1)=1, f: [0,1] --->[ 0,1]

How to prove the continuity of f ?

I think it is m(Cp)=0 and therefore

m([0,x]∩([0,1]-Cp)) = m([0,x]∩([0,x]-Cp)) = m([0,x]-Cp) = m([0,x])
and m([0,1]-Cp) = 1
Thus f(x) = m([0,x]) = x.

f = id is continuous.

Regards
Hero Wanders

If Cp is general cantor set. 0 <= m(Cp) < 1. How to prove the
continuity.
.