What are the three tests for continuity?

[UrbanXrisis]

what is the definition for discontinuity? I got a question on a math test wrong because it asked to "justify your answer" but I couldn't because I didn't know the definition for discontinuity.

 

[arildno]

Can you post your definition of CONTINUITY at a point?

 

[Diane_]

There are a couple, but the one I like is:

A function f is continuous at a point a iff lim(x -> a) f(x) = f(a).

If not, then the function is discontinuous there.

 

[arildno]

I was trying to lead up to that..
In particular, later on, it is important to specify those different cases in which continuity might fail to exist..

 

[Diane_]

I know, Arildno, and I did think about that before I posted. But since he's looking for the answer to a test question and not homework, I figured we might as well be specific. No offense intended. :)

 

[arildno]

No offense taken..

 

[UrbanXrisis]

Can you post your definition of CONTINUITY at a point?

What is the definition of continuity?

 

[marlon]

the intuitive one is that when you draw a curve between two points, you should not have to raise your hand while doing so...

Keep in mind that there is left and right continuity but they are also easy to understand.

marlon

 

[BobG]

Definition for continuity should be right in your textbook (of course, textbooks have a way of turning common sense into a foreign language).

There's three tests for continuity, which might mean a little more (those should also be in your textbook).

To be continuous at some point (we'll call it c),

f(c) has to exist. For example, if f(x) = 1/x, and 0 (one possible value for c) is inserted for x, the equation would be undetermined (i.e. c=0, f(c) does not exist).

The limit of f(x) must exist as x approaches c.

f(c) must equal the limit of f(x) as x approaches c.

So, if f(x)=1/x, then it is continuous at f(c) if c=1. 1/1 equals 1, so f(c) exists. The limit of f(x) as x approaches 1 is 1. Since f(c) and the limit of f(x) as x approaches c both equal 1, f(x) is continuous at x=1.

If c=0, then f(c) doesn't exist, the limit of f(x) as x approaches 0 doesn't exist, rendering the third test moot (and impossible to conduct, in this case). Actually, as soon as any of the tests fail, you can stop.