Download - 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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