# continuous function betweem two metric spaces

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• Nov 18th 2010, 02:53 AM
1234567
continuous function betweem two metric spaces
Hi can some please help me with the following problem:

Attachment 19755

Thanks
• Nov 18th 2010, 04:02 AM
FernandoRevilla
Hint:

Prove: $(f_n)\rightarrow f \Rightarrow (\Delta f_n)\rightarrow \Delta f$.

Regards.

Fernando Revilla
• Nov 18th 2010, 10:26 AM
Jose27
Quote:

Originally Posted by 1234567
Hi can some please help me with the following problem:

Attachment 19755

Thanks

Another thing you could do is to prove that evaluation at a point is a continous linear functional (for this it's enough to prove that the kernel of the mapping is closed, which is easy) and note that the original mapping is just a sum of these.
• Nov 18th 2010, 02:00 PM
Drexel28
Personally while the other two responses are correct I think it's easier to notice that if $\|f-g\|_{\infty}<\frac{\varepsilon}{4}$ then $\displaystyle \left|f(1)-g(1)+(f(0)-g(0))-(2f(\frac{1}{2})-g(\frac{1}{2}))\right|\leqslant |f(1)-g(1)|+|f(0)-g(0)|+2|f(\frac{1}{2})-g(\frac{1}{2})|\leqslant 4\|f-g\|_{\infty}<\varepsilon$.
• Nov 19th 2010, 12:33 PM
1234567
Yes the method that Drexel28 is far the easy way to show this, thanks ;)