Let f(x) : ]-π,π[ -->R the function defined by
f(x)= e^(1/sin(x)) if x≠0, 0 if x=0
Is it uniformly continuous in ]-π,0[ ?
#Uniform continuity
late dagger
normal skyBOT
late dagger
Heine-Cantor says that if a function is continuous in [a,b], then ot's uniformly continuous in [a,b]
I can extend by continuity the function in -π by saying f(-π)=0, but I'm having trouble with x=0...
scenic cedar
Heine-Cantor says that if a function is continuous in [a,b], then ot's uniformly...
What exactly is your problem with x = 0?
late dagger
The left and right limits as x-->0 are 0 and +infinity, so it's not continuous in 0, right? So how can I extend by continuity in 0?
(I know it's uniformly continuous already, I just don't understand why)
scenic cedar
The left and right limits as x-->0 are 0 and +infinity, so it's not continuous i...
f:[-π,0] is continuous as you can't approach 0 from the right.
late dagger
Wait why can't I? It's not as if the function doesn't exist in ]0,π[
scenic cedar
Wait why can't I? It's not as if the function doesn't exist in ]0,π[
If we restrict the domain on [-π,0] there is no value bigger than 0 in the domain.
late dagger
Yeah but the original domain was ]-π,π[, shouldn't I also consider what's over 0?
scenic cedar
No, they're asking about continuity in (-π,0). So why would I care what the function does e.g. in (π/2,π)?
Continuity is a local property
late dagger
So since the left limit as x-->0 is 0 and f(0)=0, then f(x) is continuous in 0?
What about if they asked for example between ]-π,1]?
scenic cedar
So since the left limit as x-->0 is 0 and f(0)=0, then f(x) is continuous in 0?
...
No, I wasn't saying that f is continuous in 0. But if we restrict the domain on [-π,0] it is continuous in [-π,0]. I'm not sure if I am explaining well? Do you get what I mean?
late dagger
Yeah sorry I explained wrong myself, I meant continuous in 0 if we restrict the domain mb
scenic cedar
Yes, then you are right
late dagger
My point was that if 0 was not continuous then how can I extend it by continuity, but if we restrict the domain then we can right?
scenic cedar
Yes, you are right. You can't extend it continuously on [-π,π), but you can if we just look at [-π,0]
late dagger
Ight, thx a lot👍🏻
late dagger
Yeah
How do i do that?
scenic cedar
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