
Limit of a Function
Let be a function of a real variable . We say that limit of as is , when assumes values closer and closer to , assumes values closer and closer to . We write this as,

Right Hand Limit and Left Hand Limit
can approach from directions, either from its left [] or from its right []. This gives rise to the right and left hand limits. Right hand limit is denoted by
and left hand limit is denoted by
Obviously, when both right and left hand limits are equal, is defined.

Continuity and Discontinuity
The concept continuity can be regarded as the graphical viewpoint of limits. If we can draw the graph of a function without lifting our pen from the plane of paper, we say that the function is continuous everywhere. In other words, it does not have any gaps in its graph. When it has (at least one), we say that the function is discontinuous.

Continuity at a Point
When following equalities hold, function is continuous at .
This means, existence of limit is a NECESSARY but NOT SUFFICIENT condition for the function to be continuous. The limit of the function at a point must be equal to the value of the function at that point, i.e. .

Removable Discontinuity
The discontinuity of a function at a point is said to be removable, if the limit as exists, but is not equal to , or
It can be removed by redefining the function at .

Irremovable Discontinuity
The discontinuity of a function at a point is said to be irremovable, if the limit as does not exist, or
This discontinuity cannot be removed and there is a jump in the graph of function.

Algebra of Continuity of Functions
Let and be 2 real valued functions of . Suppose they are continuous at . Then, following functions are continuous at :
1) ,
2)
3)
4) and
5)

Continuity in an Interval
A function is said to be continuous in an interval, if it is continuous at each point in that interval.

Continuity in the Domain of a Function
A function is said to be continuous in its domain, if it is continuous at each point in the domain.
Recall that domain is that set of values, for which is defined.

Continuity of Few Functions
1) Polynomial functions,
They are continuous everywhere.
2) Rational functions
If and are polynomial functions, then is continuous everywhere except at point/s where .
3) Trigonometric functions
and functions are continuous everywhere.
and are continuous everywhere except at those points when is .
and are continuous everywhere except at those points when is .
4) Exponential function
is continuous everywhere.
When , it becomes a constant function, which is also continuous everywhere.
5) Logarithmic function
is continuous for every positive real number.