Question #344377

Show that the function f defined by f(x) =1//x^2 is uniformly continuous on [5 ,∞].

1
Expert's answer
2022-05-25T09:13:00-0400

ANSWER

Let ϵ>0.\epsilon >0. By the definition, we need to show there exists δ>0\delta >0 , such that


a5,b5a\geq 5 , b\geq 5 and ab<δ|a-b|<\delta imply f(a)f(b)<ϵ|f(a) - f(b)|< \epsilon (1)

The function f(x)=1x2f(x)=\frac{1}{x^{2}} is differentiable on the set [5,)[5,\infty) . f(x)=2x3f'(x)=\frac {-2} { x^{3}} and f(x)253<1|f'(x)|\leq \frac{2}{5^{3}}<1 for all x[5,).x\in [5,\infty). Let b>a5b>a\geq 5 , according to the mean value theorem, there exists point c(a,b)c\in (a,b) such that f(b)f(a)=f(c)(ba)f(b)-f(a) =f'(c)\cdot (b-a). So, for all a,b[5,)a,b\in [5,\infty) we have

f(b)f(a)ba|f(b)-f(a)|\leq |b-a| .

Hence, if ϵ>0\epsilon >0 and δ=ϵ\delta=\epsilon , then

ab<δ=ϵf(b)f(a)ba<δ=ϵ|a-b|<\delta =\epsilon \Rightarrow |f(b)-f(a)|\leq |b-a|<\delta=\epsilon

Those, (1) is satisfied, which means that the function is uniformly continuous.


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