函数y=sinx^2在(0,∞)上不一致连续,怎么证明?

Question

函数y=sinx^2在(0,∞)上不一致连续,怎么证明?

Answer 1

对ε0 =1/2 > 0,对任意的 δ >0,取 k =1/(32πδ^2),x1= sqrt(2kπ),x2 = sqrt(2kπ+π/2) ∈(-∞,+∞),有
　|x1 - x2| = sqrt(2kπ+π/2) - sqrt(2kπ)
= (π/2)/[sqrt(2kπ+π/2) + sqrt(2kπ)]
　< (π/2)/[2*sqrt(2kπ)] =……< δ,
但
　|sin[(x1)^2] - sin[(x2)^2]| = |sin(2k π) - sin (2kπ+π/2)| = 1 > ε0,
此即证得f(x)=sin(x^2)在(‐∞,+∞)上是非一致连续.