Slight Discrepancy in Piece-wise Function Graph

[brendan_foo]

Hi there,

Only a slight problem here... My girlfriend has just taken a calc 1 quiz and she's presented this problem to me which goes against the grain of what I know - however, I could be hideously mis-informed.

She was asked to draw a graph of a piece-wise function given some definitions and limits. I shall only state the area that is causing me some intuitive grief.

$$f(0) = 2$$
$$\lim_{x\to{0^-}} f(x) = -1$$
$$\lim_{x\to{0^+}} f(x) = 1$$

If f(0) is most definitely defined at that point, why is it not its limit. By this logic, couldn't i essentially dot defined, singular points all over any arbitrary graph with DEFINITE two sided limits...This is my problem...

Any opinions?

Cheers guys...Brendan

 

[arildno]

Well, as long as you don't confuse the concepts of limits and function values, this exercise cannot possibly represent any problems.

 

[brendan_foo]

This is what I argued, but wasn't received with much gratitude.

Thankyou for this...

Cheers...Brendan

 

[arildno]

Okay, remember that in order for a one-sided limit to exist at some point, then the limiting function value of EACH SEQUENCE OF NUMBERS CONVERGING TO THAT POINT (from the one side) must equal the limiting function value associated with every other sequence.

 

[brendan_foo]

Quick other point:

It then asks...is the function left continuous, right continuous or continuous at the point x = 0. Based on the definition of continuity, that is,

$$\lim_{x\to a}f(x) = f(a)$$

Based on the two stipulated limits (as is posted above), what could I say in this case, as the limits are defined as different to the stated existing point $f(0) = 2$.

Cheers...Brendan

 

[arildno]

x=0 is a screaming discontinuity for this function

 

[brendan_foo]

Again, this is what I argued - however, as a multiple choice question as: left cont, right cont, or continuous in both senses...how is one to answer?! This is the point that I make.

 

[matt grime]

It is none of them. Just look at the definitions you have been given.

 

[brendan_foo]

I can 100% see this; but when faced with a) b) or c) and nothing in between, it seemed a little peculiar, especially with her screaming down my neck that it MUST be one of the three, even though it clearly isn't.

Thanks guys, just wanted to get some confirmation before I got on my high-horse.

Mucho-gracias!

 

[arildno]

Well, as you actually PHRASED it (without the a), b) and c)), NEITHER is a perfectly acceptable answer!

If the book clearly has stated your three only options as the ones you mentioned, I suggest you make a cozy fire of it.

 

[brendan_foo]

I quote verbatim from the cherished one... It wasn't a book exercise, it was a quiz devised by some one.

 

[arildno]

Yes, so as I said "NEITHER" is a perfectly acceptable answer.