continuity and discontinuity of functions

CONTINUITY

Continuity

A function is said to be continuous at x = a if there is no interruption in the graph of f(x) at a. Its graph is unbroken at a, and

there is no hole, jump or gap.

Continuity of a function at a point

A function is said to be continuous at a point x = a if the following three conditions are satisfied:

1. f(x) is defined, that is, exists, at x = a

2. The limit of f(x) as x approaches a exists

3. The limit of f(x) as x approaches a is equal to f(a).

Example: Discuss the continuity of f(x) = 2 – x3 at x = 1.

DISCONTINUITY

Removable Discontinuity

A function is said to have removable discontinuity at x =a, if the limit of f(x) as x approaches a exists, and is not equal to

f(a)

)(;)(lim afLLxfax

Jump Discontinuity

A function is said to have jump discontinuity at x =a, if the limit of f(x) as x approaches to a from the right is not equal to the limit of f(x) as x approaches to a from the left.

)(lim)(lim xfxfaxax

Infinite Discontinuity

A function is said to have infinite discontinuity at x =a, if the limit of f(x)

as x approaches to a is infinite.

)(lim xfax

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