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Revision Notes on Continuity A function f(x) is said to be continuous at x= a if lim_{x→a}^{-} f(x)= lim_{x→a}^{+} f(x)=f(a) Thus, unlike limits, for continuity it is essential for the function to be defined at that particular point and the limiting value of the function should be equal to f(a). The function f(x) will be discontinuous at x =a in either of the following situations: lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) exist but are not equal. lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) exist and are equal but not equal to f(a). f(a) is not defined. At least one of the limits does not exist. If you are required to comment on the continuity of a function, then you may just look for the points on the domain where the function is not defined. Some important properties of continuous functions: If the functions f(x) and g(x) are both continuous at x =a then the following results hold true: cf (x) is continuous at x =a where c is any constant. f(x) + g(x) is continuous at x = a. f(x).g(x) is continuous at x= a. f(x)/g(x) is continuous at x= a, provided g(a) ≠ 0. If a function f is continuous in (a, b), it means it is continuous at every point of (a, b). If f is continuous in [a, b] then in addition to being continuous ay every point of domain, f should also be continuous at the end points i.e. f(x) is said to be continuous in the closed interval [a, b] if f(x) is continuous in (a, b) lim_{x→a}^{+} f(x)=f(a) lim_{x→a}^{-} f(x)=f(a)
lim_{x→a}^{-} f(x)= lim_{x→a}^{+} f(x)=f(a)
Thus, unlike limits, for continuity it is essential for the function to be defined at that particular point and the limiting value of the function should be equal to f(a).
lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) exist but are not equal.
lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) exist and are equal but not equal to f(a).
f(a) is not defined.
At least one of the limits does not exist.
If the functions f(x) and g(x) are both continuous at x =a then the following results hold true:
cf (x) is continuous at x =a where c is any constant.
f(x) + g(x) is continuous at x = a.
f(x).g(x) is continuous at x= a.
f(x)/g(x) is continuous at x= a, provided g(a) ≠ 0.
If a function f is continuous in (a, b), it means it is continuous at every point of (a, b).
If f is continuous in [a, b] then in addition to being continuous ay every point of domain, f should also be continuous at the end points i.e. f(x) is said to be continuous in the closed interval [a, b] if
f(x) is continuous in (a, b)
lim_{x→a}^{+} f(x)=f(a)
lim_{x→a}^{-} f(x)=f(a)
While solving problems on continuity, one need not calculate continuity at every point, in fact the elementary knowledge of the function should be used to search the points of discontinuity.
In questions like this where a function h is defined as
h(x) = f(x) for a < x < b
g(x) for b < x < c
The functions f and g are continuous in their respective intervals, then the continuity of function h should be checked only at the point x = b as this is the only possible point of discontinuity.
If the point ‘a’ is finite, then the necessary and sufficient condition for the function f to be continuous at a is that lim_{x→a}^{-} f(x) and lim_{n→a}^{+} f(x) should exist and be equal to f(a).
A function continuous on a closed interval [a, b] is necessarily bounded if both a and b are finite. This is not true in case of open interval.
If the function u = f(x) is continuous at the point x=a, and the function y=g(u) is continuous at the point u = f(a), then the composite function y=(gof)(x)=g(f(x)) is continuous at the point x=a.
Given below is the table of some common functions along with the intervals in which they are continuous:
Functions f(x)
Interval in which f(x) is continuous
Constant C
(-∞,∞)
b^{n}, n is an integer > 0
|x-a|
x^{-n}, n is a positive integer.
(-∞,∞)-{0}
a_{0}x^{n} + a_{1}x^{n-1 }+........ + a_{n-1}x + a_{n}
p(x)/q(x), p(x) and q(x) are polynomials in x
R - {x:q(x)=0}
sin x
R
cos x
tan x
R-{nπ:n=0,±1,........}
cot x
R-{(2n-1)π/2:n=0,±1,± 2,........}
sec x
e^{x}
ln...
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Solved Examples on Continuity Illustration 1: The...