More elaborately, if the left hand limit, right hand limit and the value of the function at exist and are equal to each other, i.e.,
Then is said to be continuous at .
5.1.2 Continuity in an interval
(i) is said to be continuous in an open interval if it is continuous at every point in this interval.
(ii) is said to be continuous in the closed interval if
is continuous in
5.1.3 Geometrical meaning of continuity
(i) Function will be continuous at if there is no break in the graph of the function at the point .
(ii) In an interval, function is said to be continuous if there is no break in the graph of the function in the entire interval.
5.1.4 Discontinuity
The function will be discontinuous at in any of the following cases:
(i) exist but are not equal.
(ii) exist and are equal but not equal to .
(iii) is not defined.
5.1.5 Continuity of some of the common functions
Function | Interval in which is continuous | |
1. | The constant function, i.e. | R |
2. | The identity function, i.e. | R |
3. | The polynomial function, i.e. | R |
4. | ||
5. | is a positive integer | |
6. | , where and are polynomials in | |
7. | R | |
8. | ||
9. | ||
10. | ||
11 | ||
12 | The inverse trigonometric functions, | In their respective domains |
5.1.6 Continuity of composite functions
Let and be real valued functions such that is defined at a. If is continuous at a and f is continuous at , then is continuous at a.
5.1.7 Differentiability
The function defined by , wherever the limit exists, is defined to be the derivative of at . In other words, we say that a function is differentiable at a point in its domain if both , called left hand derivative, denoted by , and , called right hand derivative, denoted by , are finite and equal.
(i) The function is said to be differentiable in an open interval if it is differentiable at every point of
(ii) The function is said to be differentiable in the closed interval if and exist and exists for every point of .
(iii) Every differentiable function is continuous, but the converse is not true
5.1.8 Algebra of derivatives
If are functions of , then
(i)
(ii)
(iii)
5.1.9 Chain rule is a rule to differentiate composition of functions. Let . If
5.1.10 Following are some of the standard derivatives (in appropriate domains)
1.
2.
3.
4.
5.
6.