Essential discontinuity - terminology (1 Viewer)

[leehuan]

These are just when at least one left or right side limit is undefined right? So obviously sin(1/x) has this problem at x=0 due to the rapidly oscillating behaviour. Would vertical asymptotes also get classified as this?

 

[Paradoxica]

No, because they have a fixed direction and tend towards signed infinity.

i.e. the limit exists in the sense of that it has a value it tends towards, in this case, positive or negative infinity (but not both) [from a given direction]

tan(1/x), on the other hand, tends to both infinities as x approaches zero, and has an essential discontinuity.

 

[InteGrand]

These are just when at least one left or right side limit is undefined right? So obviously sin(1/x) has this problem at x=0 due to the rapidly oscillating behaviour. Would vertical asymptotes also get classified as this?

Not necessarily undefined. It basically means that there's no value you could define for the function there that would make it continuous there (as opposed to a 'removable discontinuity', which is one where you can define a function value appropriately and make the function continuous there).

Vertical asymptotes don't count as discontinuities because the x-values there aren't actually part of the function's domain (unless you happened to define the function there with some value).

Edit: I was describing jump discontinuity, sorry! Check out here: https://en.wikipedia.org/wiki/Classification_of_discontinuities .

So essential discontinuity means one or both of the left/right hand limits do not exist or are infinite.

(The point in question needs to be part of the function's domain. So yes, vertical asymptotes will count, but only if the function has a value defined there.)

 

[leehuan]

Not necessarily undefined. It basically means that there's no value you could define for the function there that would make it continuous there (as opposed to a 'removable discontinuity', which is one where you can define a function value appropriately and make the function continuous there).

Ah right.

In that case another question. Obviously I can look at a removable discontinuity and jump discontinuity and distinguish between the two. In the exam how would I describe the difference though?

Would it be that whilst f(a) neq limit, the left and right limits are equal -> removable?

 

[Paradoxica]

Not necessarily undefined. It basically means that there's no value you could define for the function there that would make it continuous there (as opposed to a 'removable discontinuity', which is one where you can define a function value appropriately and make the function continuous there).

Vertical asymptotes don't count as discontinuities because the x-values there aren't actually part of the function's domain (unless you happened to define the function there with some value).

e.g. e^(sin(1/x)) has an essential discontinuity at x = 0

 

[Paradoxica]

Ah right.

In that case another question. Obviously I can look at a removable discontinuity and jump discontinuity and distinguish between the two. In the exam how would I describe the difference though?

Would it be that whilst f(a) neq limit, the left and right limits are equal -> removable?

no, the limit doesn't even exist.

 

[leehuan]

no, the limit doesn't even exist.

Wait what, this just confused me.

I thought removable was something like this https://upload.wikimedia.org/wikipedia/commons/9/9d/Discontinuity_removable.eps.png

And here clearly left limit = right limit surely?

Edit: Well I guess maybe f(a) doesn't have to exist but that aside

 

[InteGrand]

Ah right.

In that case another question. Obviously I can look at a removable discontinuity and jump discontinuity and distinguish between the two. In the exam how would I describe the difference though?

Would it be that whilst f(a) neq limit, the left and right limits are equal -> removable?

Sorry what I said about 'not necessarily undefined' was actually in reference to 'jump discontinuity', got my terms mixed up.

To distinguish between a removable discontinuity and another discontinuity (jump or essential), the former type is one where the function can be defined at the point in such a way that the function will be continuous there then.

E.g. If f: R -> R, f(x) = (x^2 -1)/(x-1) for x =/= 1, and f(1) = 3 has a removable discontinuity at x=1. It is a discontinuity, but removable because we can redefine f(1) = 2 and then you can show that f is continuous at 1.

To show that a point is an essential discontinuity or jump discontinuity, basically show it satisfies the definition of it. E.g. to show a is an essential continuity (where a is a point in the domain of f and f is also defined in some neighbourhood of a, so that limits to a will make sense), it'd suffice to show that one of the left or right hand limits as x -> a of f(x) does not exist or is infinite.

 

[Paradoxica]

Sorry what I said about 'not necessarily undefined' was actually in reference to 'jump discontinuity', got my terms mixed up.

To distinguish between a removable discontinuity and another discontinuity (jump or essential), the former type is one where the function can be defined at the point in such a way that the function will be continuous there then.

E.g. If f: R -> R, f(x) = (x^2 -1)/(x-1) for x =/= 1, and f(1) = 3 has a removable discontinuity at x=1. It is a discontinuity, but removable because we can redefine f(1) = 2 and then you can show that f is continuous at 1.

Ok, I'll apply this with examples to distinguish the three

Removable Discontinuity : sinx / x has a removable discontinuity at 0

Essential Discontinuity : sin(1/x) has an essential discontinuity at the origin

Jump Discontinuity: x/sin(x) has (countably) infinitely many jump discontinuities at integer multiples of pi.

Now, we move to a more complicated case

xsin(1/x) has a removable discontinuity at the origin, since even though it's oscillations grow, it's magnitude vanishes.

 

[leehuan]

Alright it's all fine as to these examples so far. Lastly can someone just tell me what 1/x at x=0 is.

 

[Paradoxica]

Alright it's all fine as to these examples so far. Lastly can someone just tell me what 1/x at x=0 is.

whatever you want it to be...

unless you treat it as a jump discontinuity, then it doesn't have a value, but the limits from both sides work.

 

[InteGrand]

Alright it's all fine as to these examples so far. Lastly can someone just tell me what 1/x at x=0 is.

It's undefined. If we consider the function f: R -> R, defined as f(x) = 1/x for x =/= 0 and f(0) = 1 (say), then f will have an essential discontinuity at 0, since the limit from a side (in fact both sides here) is infinite (where infinite means limit being +/- infinity).

 

[leehuan]

It's undefined. If we consider the function f: R -> R, defined as f(x) = 1/x for x =/= 0 and f(0) = 1 (say), then f will have an essential discontinuity at 0, since the limit from a side (in fact both sides here) is infinite (where infinite means limit being +/- infinity).

Oh is this what I missed. We have to define a random value for f(0) first to call it essential (otherwise it's simply undef.)? Ok I see.

 

[InteGrand]

Oh is this what I missed. We have to define a random value for f(0) first to call it essential (otherwise it's simply undef.)? Ok I see.

Basically yeah. But even if we define it, we wouldn't say that 1/x is 1 at 0. We'd say f is 1 at 0, where f(x) is only 1/x away from x = 0. 1/x itself would always be undefined at x = 0 because we can't divide by 0.

 

[leehuan]

Ok I see. Thanks I know what I forgot now