Question #170272

Assume that. Fn converges to f uniformly on S if each fn is continuous at a point c of S .show that the limit function f us also continuous at c


1
Expert's answer
2021-03-10T12:07:56-0500

Solution:

Since fnff_n → f uniformly on S and each fnf_n is continuous on S, for  given ε>0ε > 0

there is a δ>0\delta>0 such that as yx<δ|y-x|<\delta , where ySy \in S , we have

f(y)f(x)<ε2|f(y)-f(x)|<\frac{\varepsilon}{2}

For this δ>0,\delta>0,\exists a positive integer N1N_{1} such that as nN1n \geq N_{1} , we have

xnx<δ\left|x_{n}-x\right|<\delta

Hence, as nN1n \geq N_{1} , we have

f(xn)f(x)<ε2 ...(1)\left|f\left(x_{n}\right)-f(x)\right|<\frac{\varepsilon}{2}\ ...(1)

Also, since fnff_{n} \rightarrow f uniformly on S, given ε>0,\varepsilon>0,\exists a positive integer NN1N \geq N_{1} such that as nNn \geq N , we have

fn(x)f(x)<ε2 for all xS\left|f_{n}(x)-f(x)\right|<\frac{\varepsilon}{2} \text { for all } x \in S

fn(xn)f(xn)<ε2 ...(2)\Rightarrow \left|f_{n}\left(x_{n}\right)-f\left(x_{n}\right)\right|<\frac{\varepsilon}{2} \ ... (2)

By (1) and (2), we obtain that given ε>0,\varepsilon>0,\exists a positive integer N such that as nNn \geq N , we have

fn(xn)f(x)=fn(xn)f(xn)+f(xn)f(x)<ε2+ε2=ε\begin{aligned} \left|f_{n}\left(x_{n}\right)-f(x)\right| &=\left|f_{n}\left(x_{n}\right)-f\left(x_{n}\right)\right|+\left|f\left(x_{n}\right)-f(x)\right| \\ &<\frac{\varepsilon}{2}+\frac{\varepsilon}{2} \\ &=\varepsilon \end{aligned}

fn(xn)f(x)\Rightarrow f_{n}\left(x_{n}\right) \rightarrow f(x)

Hence, limit function f is also continuous at any point c.

Hence, proved.


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Comments

Assignment Expert
13.03.21, 00:24

Dear Prathibha, You are welcome. We are glad to be helpful. If you liked our service, please press a like-button beside the answer field. Thank you!

Prathibha
12.03.21, 16:51

Thanks....

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