Math 22 Continuity

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Informally, a function is continuous at means that there is no interruption in the graph of at .

Definition of Continuity

 Let  be a real number in the interval , and let  be a function whose domain contains the interval  . The function  is continuous at  when 
 these conditions are true.
 1.  is defined.
 2.  exists.
 If  is continuous at every point in the interval , then  is continuous on the open interval .

Continuity of piece-wise functions

Discuss the continuity of

On the interval , and it is a polynomial function so it is continuous on
On the interval , and it is a polynomial function so it is continuous on
Finally we need to check if is continuous at .
So, consider
Then, .
Since , \lim_{x\to 3} f(x) exists.
Also notice
So by definition of continuity, is continuous at .
Hence, is continuous on

Types of Discontinuity

Removable discontinuity: If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it. For example: . This function is y=x-3 with a hole at since makes undefined.

Infinite discontinuity: An infinite discontinuity exists when one of the one-sided limits of the function is infinite and the limit does not exist. This is an infinite discontinuity. In another word, we have infinite discontinuity when either or

Jump discontinuity: The function is approaching different values depending on the direction is coming from. When this happens, we say the function has a jump discontinuity at . In another word,


Polynomial function is continuous on the entire real number line (ex: is continuous on )

Rational functions is continuous at every number in its domain. (ex: is continuous on since the denominator cannot equal to zero)

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