Elementary Problem: Determining the Piecewise Form of a Function

[auslmar]

Hello All,

First off, I will apologize in advance for being so ignorant and scatterbrained. Please humor me.

In my Calc. I class, we are currently covering the continuity of functions. Our current problem set consists of determining where a given function is NOT continuous. My professor insists that we must find the piecewise form of the functions, analyze that, and then determine the continuity.

I'm stumped on a couple of points:

I can't seem to figure out exactly how to determine the piecewise form of any given function to even start. Is there a certain method or rule to finding this? I can, however, understand how to determine the piecewise form of of a function that is blatantly co-linear by analyzing the graph, noting the branching point(s), and finding the slope of the line(s) on either side of the branching point(s). But, as far as determining the piecewise form of something like (x)/(x-2), I'm lost.

Also, when considering the continuity of functions like these, I don't see why we couldn't just consider the natural domain of the function and test the continuity at the points of interest. Is that way of thinking about these problems flawed?

And finally, I have a very very stupid question. When considering if f(c) exists and the limit of f(x) as x approaches c exists, what if they're both non-existent? Are two non-existences equal? That's probably a very ignorant question, but I'm curious.

I'm sorry this post turned into an incoherent ramble. But, if you can help me out in any way, I'd be most appreciative.

Thanks for your consideration and patience,

-A.Martin

 

[Gib Z]

I don't see why your only consider the Natural domain either, normally its all the real numbers, not just the natural numbers. Its very flawed. Only considering the natural numbers, a function may be continuous in its domain, but not able to be differentiated.

For \lim_{x\rightarrow c} f(x) to exist, \lim_{x\rightarrow c^{+}} f(x) and \lim_{x\rightarrow c^{-}} f(x) both have to exist and be equal. If the limit does not exist, f(c) can not exist. If the limit exists, it does not mean f(c) exists.

 

[auslmar]

I don't see why your only consider the Natural domain either, normally its all the real numbers, not just the natural numbers. Its very flawed. Only considering the natural numbers, a function may be continuous in its domain, but not able to be differentiated.

For \lim_{x\rightarrow c} f(x) to exist, \lim_{x\rightarrow c^{+}} f(x) and \lim_{x\rightarrow c^{-}} f(x) both have to exist and be equal. If the limit does not exist, f(c) can not exist. If the limit exists, it does not mean f(c) exists.

Okay, I see what you're saying.

What about determining the piecewise form of a function? Is there any systematic way to go about that?

 

[Gib Z]

Nope. Not all functions can be expressed piecewise anyway, unless in a very obvious way that doesn't help. eg f(x) { for x>0, =x^2, for x< 0 = x^2. Id be interesting to see how you espressed the x/x-2, i can't do it.