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Continuity

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
 
Continuous function is a function whose graph can be drawn without lifting the pen from the paper.

Definition of continuity of a function

A function f(x) is said to be continuous at x = aif 88263.png = 88257.png = f(a), i.e., LHL = RHL = value of a function at x = a or 88251.png = f(a).
 
A function f(x) is said to be discontinuous at x = a, if
  1. 88245.png and 88239.png exist but are not equal.
  2. 88231.png and 88225.png exist and are equal but not equal to f(a).
  3. f(a) is not defined.
  4. At least one of the limits does not exist.
Note: of a function is the property of interval and is meaningful at x = a only if the function has a graph in the immediate neighborhood of x = a, not necessarily at x = a. Hence it should not be mislead that continuity of a function is talked only in its domain.
 
For example, continuity of f(x) = 89081.png at x = 1 is meaningful but continuity of f(x) = loge x at x = –2 is meaningless. Similarly, if f(x) has a graph as shown then continuity at x = 0 is meaningless.
 
88908.png

Also continuity at
x = a ⇒ existence of limit at x = a but existence of limit at x = a does not mean continuity at x = a.

Directional continuity

A function may happen to be continuous in only one direction, either from the “left” or from the “right”.
 
A right-continuous function is a function which is continuous at all points when approached from the positive infinity.
 
Likewise a left-continuous function is a function which is continuous at all points when approached from the negative infinity.
 
88168.png

Continuity in interval

A function is said to be continuous in the open interval (a, b) if f(x) is continuous at each and every point (a, b). For any c (a, b) 88162.png = 88156.png = f(c).
 
A function f(x) is said to be continuous in the closed interval [a, b] if it is continuous at every point in this interval and the continuity at the end points is defined as f(x) is continuous at x = a if f(a) = 88150.png = RHL (LHL should not be evaluated) and at x = b if f(b) = 88144.png = LHL (RHL should not be evaluated).




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