MATH1051 Chapter Notes - Chapter 5: Negative Number, Intermediate Value Theorem, Bisection Method
Chapter 5 - Continuity
5.1 Definition: Continuity
We say that a function is continuous at a point if:
• exists
• The limit at exists
•
Picture a graph of the function. The first criterion ensures there’s a point somewhere on the
graph when . The second criterion ensures that there are lines that seem to meet up
when
(though, at this stage, there could be a break!). The third criterion ensures the lines
meet up with that little dot.
If any of these criteria fail, the function isn’t continuous at that point .
5.2 Continuity on Intervals
If is continuous on an open interval , then is continuous at all points where .
If is continuous on a closed interval , then must meet the above criteria, and the value
at each point must be equal to the limit you get when approaching that point
from inside the interval .
A continuous function is something you can draw in one pen stroke.
Any polynomial in is continuous in
is continuous on .
5.3 Properties
If both and are continuous at and is a constant, then the following are all
also continuous:
•
•
•
•
Basically, if you have two continuous functions, combining them with a simple operation
will also yield a continuous function.
If is continuous at , and the limit of as you approach is … then the limit as you
approach of the composite function is simply . Applying to at that point is
the same as
Document Summary
We say that a function is continuous at a point if: exists: the limit at exists graph when when. The second criterion ensures that there are lines that seem to meet up. The first criterion ensures there"s a point somewhere on the. If any of these criteria fail, the function isn"t continuous at that point (though, at this stage, there could be a break!). The third criterion ensures the lines meet up with that little dot. , then is continuous at all points where. , then must meet the above criteria, and the value at each point from inside the interval must be equal to the limit you get when approaching that point. A continuous function is something you can draw in one pen stroke. Any polynomial in is continuous in is continuous on are continuous at and is a constant, then the following are all.
