Continuous	A function is continuous at a number a if  Three things must take place for this to be true: 1) a must be in the domain of f, 2)  exists, and  3)
Continuous from the left	A function is continuous from the left if  (This is called a left-hand limit.
Continuous from the right	A function is continuous from the right if (This is called a right-hand limit.
Continous on an interval	A function is continuous on an interval if it is continuous at every point of the interval. At the endpoints of the interval, the function must be continuous from right (at the right end) and continuous from the left (at the left end).

We shall now examine some different types of disocntinuity.
 

Is  continuous at x = 3?	We can find out by taking the limit of the function as x approaches 3.
	We can easily find the limit.
but	Hence f(x) is not continuous or discontinuous at x = 3. This type of discontinuity is called removable discontinuity because we could remove the discontinuity by redefining  f  at the point of discontinuity. This could be done by redefining function  f   in this way:

Is  continuous at x = 0?	We can find out by taking the limit of the function as x approaches 0.
	So , so this function is discontinuous at x = 0.  This type of discontinuity is called infinite discontinuity.

Is  continuous at x = 2?	We can find out by taking the limit of the function as x approaches 2.
	This is the left-hand limit.
	This is the right-hand limit.
	This is because the left-hand limit does not agree with the right-hand limit. This is called a jump discontinuity.

Here is a visual interpretation of the three different types of discontinuity:

Here we have a linear function with a hole in it. This is removable discontinuity.	This is an example of infinite discontinuity.	This is an example of a jump discontinuity.	This is another example of a jump discontinuity.

Continuity

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