Continuity
Definition of Continuity
A function f(x) is defined for x = a is said to be continuous at x = a if:
1. f(a), i.e., the value of f(x) at x = a is a definite number and
2. The limit of the function f(x) as x → a exists and is equal to the value of f(x) at x = a.
Note: On comparing the definitions of limit and continuity we find that a function f(x) is continuous at x=a if
Limit f(x)=f(a)
x→a
Thus f(x) is continuous at x = a if we have f(a + 0) = f(a – 0) = f(a), otherwise it is discontinuous at x = a.
Continuity in an Open Interval
A function f is said to be continuous in the open interval (a, b), if it is continuous at each point of open interval.
Continuity in a Closed Interval
Let f be a function defined on the closed interval (a, b) f is said to be continuous on the closed interval [a, b] if it is:
1. continuous from the right at a and
2. continuous from the left at b and
3. continuous on the open interval (a, b).
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