Section 2.5: Continuity

Okay...so, it's my turn to write-up the blog. I have no idea what I'm doing, but I'll definitely try. So, here goes nothing. ^^

So, this section is (obviously) about continuity and discontinuity.

The book's definition of continuity is this:

A function f is continuous at a number a if


Now, this definition implies 3 required conditions:

1) There is a Limit: lim_{x® a} f(x) exists

2)We do have a value at f(a)

and...

3) lim_{x® a} f(x) = f(a)

Now, if any function violates any of these conditions, then that function is said to be "discontinuous."

Here are some examples of some discontinuous functions:











All functions (polynomial, rational, root, trigonometric, inverse trigonometric, exponential, and logarithmic) are continuous within their domains.

This means that certain functions, such as power functions, exponential functions, sin functions, cosine functions, and polynomial functions, are ALWAYS continuous because they have domains of all real numbers.


The Intermediate Value Theorem:
"Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a,b) such that f(c) = N."



If N exists, there also needs to be a 'c' value to get you to N from 'c'.

Then there exists a value 'c' between a and b such that f(c) = N

The 'c' needs to be between the a and b in order for the graph to be continuous (this is a condition for the intermediate value theorem).

Note: The intermediate value theorem is also used in locating roots of equations.



Well, here are some simple example problems:


Is continous at x = 3? Answer: No, the function is not continuous at x=3 because 3 is out of the function's domain.



Is
continous at x = 0? Answer: No, because the limit of g(x) as x approaches 0, does not exist.

If you're having some trouble with these concepts, visit:
Here

or

Here

Well, there you have it. I've finally finished this blog...now to have some fun!

...Oh yea, naljean...lucky for you! You're up next!


Haha...this one below just MIGHT happen to us!