Question

若f(x)在[0,1]上连续,求F(x)=f(x)-f(x+1/n)的连续区间.

Answer 1

连续函数之差仍为连续函数

f(x)在[0,1]上连续,

若n>0,

f(x+1/n)在[-1/n,1-1/n]上连续,

[0,1]∩[-1/n,1-1/n]=[0,1-1/n]

F(x)=f(x)-f(x+1/n)的连续区间为[0,1-1/n]