how do one determine continuty or discontinuity? thanks

Question

anonymous · Answer

attachment

anonymous · Answer

for continous function 

In general, we say that the function f is continuous at some point c of its domain if, and only if, the following holds:
The limit of f(x) as x approaches c through domain of f does exist and is equal to f(c); in mathematical notation,. If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c through values not equal c. Thus, for example, every function whose domain is the set of all integers is continuous.

anonymous · Answer

??

anonymous · Answer

$$\lim_{x \rightarrow c}f(x)=f(c)$$

anonymous · Answer

this is the simple definition in terms of limit

anonymous · Answer

epsilon delta....yuk.

anonymous · Answer

Uniform is better than pointwise perhaps...

anonymous · Answer

On an intuitive level, we can determine the continuity of discontinuity of a function by looking at its graph.  If it can be drawn without lifting the pencil, the graph is a continuous curve, and the function is continuous.  If the pencil has to be lifted in order to draw the graph, then the function is discontinuous.
   Limitwise, if a function f(x) is continuous at x = a, then $$a  \in Dom(f),$$$$\lim_{x \rightarrow a}$$exists, and$$\lim_{x \rightarrow a}f(x) = f(a).$$

anonymous · Answer

Hmmm.....

anonymous · Answer

When you say you want to "determine" continuity, what do you mean exactly?

Just look at some random function and see whether it is continuous at a point (or at all)?

anonymous · Answer

Consider a polynomial.  A theorem states that a polynomial is continuous for all real numbers.

anonymous · Answer

It should be ftraightforward to prove for a linear function a x + b.

anonymous · Answer

is continuous.

anonymous · Answer

Please pardon my keyboarding.  My fingers get too fat sometimes.

anonymous · Answer

I would like to know if the "engineer" is referring to such things.

anonymous · Answer

I ssah Hmmmm...

anonymous · Answer

Who is the "engineer"?

anonymous · Answer

engineer zeey, the op

anonymous · Answer

The guy that runs this place?

anonymous · Answer

Does he? Says he is a neophyte, for whatever that's worth..

anonymous · Answer

I just joined last month, and I'm still learning my way around here.

anonymous · Answer

Me too, I meant the guy (gal) who asked the original question.

anonymous · Answer

About continuity?

anonymous · Answer

Course, he may never return and we're all just taking to ourselves:-)

anonymous · Answer

Scrolling up...  I see.  Well, if nothing else, it was a fair to decent mental workout.

anonymous · Answer

Yes:  "how do one determine continuty or discontinuity? thanks"

I would like to know what he means by 'determine' and whether or not he is referring to elementary functions.

anonymous · Answer

to determine = to find out.

anonymous · Answer

I was wondering whether he meant "prove".

anonymous · Answer

Not necessarily elementary functions.  There are many nomelementary functions that are continuous, and some of them are contimuous for all real numbers.

anonymous · Answer

Or whether simply a list of techniques would suffice.

anonymous · Answer

There are even more (cardinality aside) noncontinuous functions.

anonymous · Answer

If the engineer means to prove continuity of a function, then he most likely will be doing an epsilon=delta proof, based on the definition of the limit.

anonymous · Answer

Which was my first contribution, in hopes of eliciting a reponse, to no avail.

anonymous · Answer

Drat:^(

anonymous · Answer

Anyway, he has disappeared, so I will see u anon...nice chat.

anonymous · Answer

:^)

anonymous · Answer

tanks