Continuity

Functions
A function which is a continuous graph, is a continuous function. In case of a disruption, the function is discontinuous.

For a function to be continuous at a point, the value of the function at that point should be equal to the limiting value. For a function to be discontinuous at a point, the value of the function at that point should be different from the limiting value.

Causes of Discontinuity
1) Limit does not exist . ( this has 3 reasons refer Limits )

2) The value of function at that point is different than the limiting value (if limit exists)

Continuity of a Function at a point
The limiting value is obtained by evaluating the limits, and the actual value of the function is obtained by substituting directly in the given equation. We need to check the limiting value and the actual value of the function to decide its continuity. If both the values are same at a given point, then the function is continuous at that particular point.

The continuity of a function at the end point of a graph is checked by considering only one side from where the graph is approaching the point. e.g. if the graph begins from x = 2 ; then the limiting value for a point x = 2, will be its Right hand limit , since the point is being approached from the right hand side.

Continuity of a function in a given Interval
An Interval can be Open or Closed. An open interval is defined from A to B, excluding A and B (a,b). A closed interval is defined from A to B, including A and B [a,b].

If the interval is open, (a,b) , we need to check continuity at all points. If the interval is closed, [a,b] , we need to check continuity at all points including the end points.

Types of Discontinuity
1) When Limit Exists (Removable Discontinuity)

2) When Limit does not Exist (Non Removable Discontinuity)

When limit exists in a discontinuous function, the function can be made continuous by simply redefining the function , where functional value = limiting value.