Continuity
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The document discusses continuity and discontinuity of functions. It defines a function as continuous at a point if it satisfies three conditions: 1) it is defined at that point, 2) the limit as it approaches the point exists, and 3) the limit equals the function value at that point. It gives an example of checking continuity at a point. It then defines three types of discontinuity: removable discontinuity where the limit does not equal the function value, jump discontinuity where the left and right limits do not match, and infinite discontinuity where the limit is infinite.
Continuity
- 1. CONTINUITY
- 2. 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.
- 3. 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).
- 4. Example: Discuss the continuity of f(x) = 2 – x3 at x = 1.
- 6. 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)
- 7. lim f ( x) L; L f (a) x a
- 8. 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.
- 9. lim f ( x) lim f ( x) x a x a
- 10. 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.
- 11. lim f ( x) x a
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