Limits and Continuity

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Limits and Continuity
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Intro to Continuity
As we have seen some graphs have holes in them,
some have breaks and some have other
irregularities. We wish to study each of these
oddities.
We will use our information of limits to decide
if a function is continuous or has holes.
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Continuity

• Intuitively, a function is said to be continuous
  if we can draw a graph of the function with one
  continuous line. I. e. without removing our
  pencil from the graph paper.

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Definition
A function f is continuous at a point x c if
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Example
f (x) x 1 at x 2.
1
2
Therefore the function is continuous at x 2.
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Example
f (x) (x2 9)/(x 3) at x -3
-3
The limit exist!
-6
Therefore the function is not continuous at x
-3.
You can use table on your calculator to verify
this.
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Continuity Properties
If two functions are continuous on the same
interval, then their sum, difference, product,
and quotient are continuous on the same interval
except for values of x that make the denominator
0.
Every polynomial function is continuous.
Every rational function is continuous except
where the denominator is zero.
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Continuity Summary.
Functions have three types of discontinuity.
Consider -
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Worksheet L.4-1