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**A function f(x) is said to be continuous at x= a if**

lim_{xâ†’a}^{-}f(x)= lim_{xâ†’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:

1. lim_{xâ†’a}^{-}f(x) and lim_{xâ†’a}^{+}f(x) exist but are not equal.

2. lim_{xâ†’a}^{-}f(x) and lim_{xâ†’a}^{+}f(x) exist and are equal but not equal to f(a).

3. f(a) is not defined.

4. 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:

1. cf (x) is continuous at x =a where c is any constant.

2. f(x) + g(x) is continuous at x = a.

3. f(x).g(x) is continuous at x= a.

4. f(x)/g(x) is continuous at x= a, provided g(a) â‰ 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

1. f(x) is continuous in (a, b)

2. lim_{xâ†’a}^{+}f(x)=f(a)

lim_{xâ†’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 â€˜aâ€™ is finite, then the necessary and sufficient condition for the function f to be continuous at a is that lim
_{xâ†’a}^{-}f(x) and lim_{nâ†’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 continuousConstant C(-âˆž,âˆž)bn, n is an integer > 0(-âˆž,âˆž)|x-a|(-âˆž,âˆž)x
^{-n}, n is a positive integer.(-âˆž,âˆž)-{0}a_{0}x^{n}+ a_{1}x^{n-1 }+........ + a_{n-1}x + a_{n}(-âˆž,âˆž)p(x)/q(x), p(x) and q(x) are polynomials in xR - {x:q(x)=0}sin xRcos xRtan xR-{nÏ€:n=0,Â±1,........}cot xR-{(2n-1)Ï€/2:n=0,Â±1,Â± 2,........}sec xR-{(2n-1)Ï€/2:n=0,Â±1,Â± 2,........}e^{x}Rln x(0, âˆž)

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
_{xâ†’a}^{- }f(x) = L_{1}and lim_{xâ†’a}^{+}f(x) = L_{2}, where L_{1}and L_{2 }are 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
_{xâ†’a}^{+ }f(x) nor lim_{xâ†’a}^{-}f(x) exist. - If any one of lim
_{xâ†’a}^{+}f(x) or lim_{xâ†’a}^{-}f(x) exists and the other does not then the function f is said to have mixed discontinuity. - If lim
_{xâ†’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
_{xâ†’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
_{xâ†’a}^{-}f(x) â‰ lim_{xâ†’a}^{+}f(x) and f(x) and may be equal to either of previous limits.

The concepts of limit and continuity are closely related. Whether a function is continuous or not can be determined by the limit of the function.

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