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## Main Question or Discussion Point

Hey guys, I'm in a little bit of a jam here:

I managed to miss a really important lecture on continuity the other day, and there were a few examples that the professor provided to the class that I just got, but would love it if someone could explain them to me.

First,

Second, we used the fact that "a function [tex]f:A \rightarrow[/tex]

[tex]\left| x_n - y_n\right| \rightarrow 0 [/tex] but [tex]\left| f(x_n) - f(y_n)\right| \rightarrow \epsilon_0 [/tex]

to show

Thirdly, we showed that

I managed to miss a really important lecture on continuity the other day, and there were a few examples that the professor provided to the class that I just got, but would love it if someone could explain them to me.

First,

*f(x)=x*is continuous on all of^{3}**R**. Everyone went through this, but I'm not really sure how it works.Second, we used the fact that "a function [tex]f:A \rightarrow[/tex]

**R**fails to be uniformly continuous on A if and only if there exists a particular [tex]\epsilon_{0} > 0[/tex] and two sequences (x_{n}) and (y_{n}) in A satisfying[tex]\left| x_n - y_n\right| \rightarrow 0 [/tex] but [tex]\left| f(x_n) - f(y_n)\right| \rightarrow \epsilon_0 [/tex]

to show

*f*is not uniformly continuous on**R**.Thirdly, we showed that

*f*is uniformly continuous on any bounded subset of**R**