- #1
VreemdeGozer
- 12
- 0
I'm taking my first course in Analysis, and we learned a couple of theorems about Uniform Continuity. I have been able to visualize most of what's been going on before, but I need some help with the following:
E [itex]\subseteq[/itex] ℝ, f: E [itex]\rightarrow[/itex] ℝ uniform continuous. if a sequence xn is Cauchy [itex]\Rightarrow[/itex] f(xn) is Cauchy
I is a closed, bounded interval, f: I [itex]\rightarrow[/itex] ℝ. if f is continuous on I [itex]\Rightarrow[/itex] f is uniformly continuous on I
We are using the international version of: An Introduction to Analysis by William R. Wade, fourth edition.
I'm really looking for a visual explanation, but if anyone can explain why it works in words, that's fine too.
E [itex]\subseteq[/itex] ℝ, f: E [itex]\rightarrow[/itex] ℝ uniform continuous. if a sequence xn is Cauchy [itex]\Rightarrow[/itex] f(xn) is Cauchy
I is a closed, bounded interval, f: I [itex]\rightarrow[/itex] ℝ. if f is continuous on I [itex]\Rightarrow[/itex] f is uniformly continuous on I
We are using the international version of: An Introduction to Analysis by William R. Wade, fourth edition.
I'm really looking for a visual explanation, but if anyone can explain why it works in words, that's fine too.