Continuity and differentiability

[knotty abyss]

sgn function is discontinuous at x = 1,2
and the GIF function should be discontinuous at all integer values of x
so total should be 0,1,2,3 4
total 5. my book says 4...because they don't account for 0. why?

[zinc basin]

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[fleet thunder]

Why is the floor function discontinuous at integers?

What is the definition of continuity?

[knotty abyss]

for example let's take [x]
now let's take an integer k
if x approaches k from left.. it will be a value less than k. so the GIF function will give (k-1)

and if it approaches k from right.. it will be greater than k.. then it GIF function would return k

with the same logic.. all gif functions are discontinuous at integers.. because LHL not equal to RHL

[fleet thunder]

What is the definition of continuity?

[knotty abyss]

lhl not equal to rhl
lhl equal to rhl but the function isn't defined at that point

there are many ways of defining continuity

[fleet thunder]

...that's not the definition of continuity.

No, there's one way.

[knotty abyss]

but I think the best way to visualize it is like if we trace the graph of a function at a point like that we'd need to uh

[fleet thunder]

A function f is continuous at a point a if and only if $\lim_{x \to a} f(x) = f(a)$.

[minor portal]

Techie Literate

[knotty abyss]

LHL = RHL = function at that point

yeah sorry I thought you meant discontinuity. my bad

[fleet thunder]

Discontinuity is literally just the absence of continuity.

[knotty abyss]

yeah

[fleet thunder]

A function f is discontinuous at a point a if and only if $\lim_{x \to a} f(x) \neq f(a)$.

[minor portal]

Techie Literate

[knotty abyss]

okay

[knotty abyss]

could you tell me why I should not consider the point 0 in this case? is 0 not an integer?

[fleet thunder]

...because it's the endpoint of the domain.

The left-sided limit doesn't exist because there's no domain of the function to the left.

The function is undefined there.

[knotty abyss]

yeah so if the limit (LHL) doesn't exist can't we say it's discontinuous

[fleet thunder]

By that logic f(x) = x^2 on [2, 4] is discontinuous at 2 and 4.

[knotty abyss]

mhm you do make sense

the thing is

I encountered a question literally before this one

have a look

[knotty abyss]

okay wait a minute before I come to that. if it's incorrect to consider 0. why are we considering 4? RHL at 4 doesn't exist too cuz it's outside the domain

[fleet thunder]

When we have an endpoint of the domain, we take the limit on the side where the domain exists.

[knotty abyss]

ah

[fleet thunder]

And $\lim_{x \to 4^-} f(x) \neq f(4)$.

[minor portal]

Techie Literate

[knotty abyss]

this one isn't really clear to me. I mean.. why don't we?

[fleet thunder]

...what?

[knotty abyss]

why do we do this

what's the logic

this is the "previous problem" I was talking about. see.. in this one they considered 0..

[fleet thunder]

Because "continuity" isn't some arbitrary thing we made up for no reason. We use it to talk about and think about the behavior of functions. "This function ends at the endpoint of its domain" is a trivial observation that doesn't have anything to do with the behavior we're talking about when we talk about continuity.

[knotty abyss]

the left hand side of 0 also lies outside the domain of this function eifht here

[fleet thunder]

What's sgn(0)?

[knotty abyss]

0

[fleet thunder]

And what's $\lim_{x \to 0^+} sgn(x)$?

[minor portal]

Techie Literate

[knotty abyss]

I know sgn is discontinuous at 0 because because RHL would give 1 and LHL would give -1

[fleet thunder]

No.

Sgn is discontinuous at 0 because sgn(0) = 0 and the limit as x approaches 0 of sgn(x) is not 0.

[knotty abyss]

yeah literally RHL not equal to LHL. that's enough to say it's discontinuous which I already said here. but yes since value of function AT 0 is neither LHL nor RHL its neither left cont nor right cont

[fleet thunder]

No.

RHL not equal to LHL is only relevant ever because it means the limit doesn't exist, therefore it's not equal to anything, therefore it's not equal to the function at the point.

[knotty abyss]

okay fine

[fleet thunder]

Again, this is the definition of continuity.

You're going to have a bad time in math if you struggle to think about definitions.

[knotty abyss]

could you please explain why we considered 0 in this question ... 0 is outside the domain the lhl part

[knotty abyss]

right. will be careful from now on

[fleet thunder]

No it's not. The statement "0 is outside the domain the lhl part" is literally meaningless.

[knotty abyss]

the left hand side of 0 doesn't lie inside the domain

could you please explain why we considered 0 in this example and not the one I sent before (the original questn)

[fleet thunder]

I already did.

The fact that the domain does not extend to the left of the point under consideration is not relevant.

And I explained why.

And I explained why we decided to do it that way.

[knotty abyss]

so why did we consider the point 0 in the sgn problem. the left of the point 0 is outside the domain of the given problem. it isn't supposed to be relevant

domain of the function doesn't extend to the left of 0 for both the problems. why are we considering it for one and not for the other

[fleet thunder]

We consider the point in both cases.

[knotty abyss]

so what's the answer to the sgn(sinx) problem?

and what's the answer to the sgn(quadratic) + [x+3] problem?

[fleet thunder]

The answers are correct.

And I have already said why.

You haven't been listening.

[knotty abyss]

this one is 5 (according to book)
the book considered 0. said function is discontinuous at x = 0

[knotty abyss]

and in this one the book says 4. there was no mention of the point 0

[knotty abyss]

you said we considered 0 for both the cases. but if we did.. then like sgn(sinx) is discontinuous for 0. it's left hand limit DOES not exist because it lies outside the domain. the function is not defined at those points.

similarly for the second problem.. the left hand part lies OUTSIDE the provided domain.

[fleet thunder]

No. Wrong.

And I've already said why it's wrong.

And you didn't listen.

[knotty abyss]

we considered 0 for sgn(sinx) and said it's discontinuous there. right?

[fleet thunder]

Stopp.

[knotty abyss]

alright

[fleet thunder]

Stop repeating the question I've already answered

Especially since you didn't listen to the answer the first time.

Answering it again is just a bigger waste of my time.

[knotty abyss]

thus one?

[fleet thunder]

Yes.

[knotty abyss]

so?

same case for both questions

[fleet thunder]

No it's not.

[knotty abyss]

how

[fleet thunder]

And I've already explained why.

I will consider repeating myself if you apologize for not listening.

[knotty abyss]

this?

[fleet thunder]

...close?

[knotty abyss]

okay what does this even mean?

[fleet thunder]

What does it look like it means?

[knotty abyss]

I don't get it. could you please explain for both the cases

[fleet thunder]

I've said the conditions under which I will consider repeating myself.

I will consider repeating myself if you apologize for not listening.

@knotty abyss

[knotty abyss]

if c is an end point only RHL for left endpoint and LHL for right end point is supposed to be checked for continuity, yes?

and now for the f(x) = sgn(sinx) problem... at 0 (0 is a left endpoint)... the right hand limit
didn't match the value of the function at that point so its discontinuous

and for the sgn(quadratic) + [x-3]
that [x-3] part
the right hand limit of the function perfectly matches the value of the function at 0 (left endpoint so right endpoint is being checked)

this is why it's continuous at 0

[fleet thunder]

I will consider repeating myself if you apologize for not listening.

[knotty abyss]

could you kindly confirm if this conclusion is correct or not?

[fleet thunder]

I will consider repeating myself if you apologize for not listening.

[knotty abyss]

you don't need to repeat everything from scratch.

I thought about this myself.. and if this is wrong please say it. I'll go through it again

[fleet thunder]

Why should I say anything to someone who doesn't listen?

[knotty abyss]

+close

[random osprey]

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