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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) Continuity in interval (a, b)

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	(-∞,∞)
bⁿ, n is an integer > 0	(-∞,∞)
\|x-a\|	(-∞,∞)
x^-n, n is a positive integer.	(-∞,∞)-{0}
a₀xⁿ + a₁x^n-1+........ + a_n-1x + 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	R
tan x	R-{nπ:n=0,±1,........}
cot x	R-{(2n-1)π/2:n=0,±1,± 2,........}
sec x	R-{(2n-1)π/2:n=0,±1,± 2,........}
e^x	R
ln x	(0, ∞)

If you know the graph of a function, it can be easily judged without even solving whether a function is continuous or not. The graph below clearly shows that the function is discontinuous.

If lim_x→a^- f(x) = L₁ and lim_x→a⁺ f(x) = L₂, where L₁and L₂are both finite numbers then it is called discontinuity of first kind or ordinary discontinuity.
A function is said to have discontinuity of second kindif neither lim_x→a⁺ f(x) nor lim_x→a^- f(x) exist.
If any one of lim_x→a⁺ f(x) or lim_x→a^- f(x) exists and the other does not then the function f is said to have mixed discontinuity.
If lim_x→a f(x) exists but is not equal to f(a), then f(x) has removable discontinuity at x = a and it can be removed by redefining f(x) at x=a.
If lim_x→a f(x) does not exist, then we can remove this discontinuity so that it becomes a non-removable or essential discontinuity.
A function f(x) is said to have a jump discontinuity at a point x=a if, lim_x→a^-f(x) ≠ lim_x→a⁺ f(x) and f(x) and may be equal to either of previous limits.