106 381
Assignments Done
94.7%
Successfully Done
In June 2022

# Answer to Question #344377 in Real Analysis for Mr Alex

Question #344377

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

1
2022-05-25T09:13:00-0400

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

"a\\geq 5 , b\\geq 5" and "|a-b|<\\delta" imply "|f(a) - f(b)|< \\epsilon" (1)

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

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

Hence, if "\\epsilon >0" and "\\delta=\\epsilon" , then

"|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.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!