Continuity and Discontinuity of Functions
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1. A function is continuous at a point if it satisfies three conditions: it is defined at that point, the limit of the function as it approaches the point exists, and the limit equals the value of the function at that point. 2. There are three types of discontinuities: removable discontinuity where the limit exists but does not equal the function value, jump discontinuity where the left and right limits do not match, and infinite discontinuity where the limit is infinity. 3. The document provides examples and explanations of continuity and different types of discontinuities in functions. It encourages the reader to check additional video resources for more information.
Continuity and Discontinuity of Functions
- 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
- 11. lim f ( x ) = ∞ x →a
- 12. Check out the videos in our class portal blog for more… www.mathonlinelearning.blogspot.com
- 13. THANK YOU AND GOODBYE! I really had a great time with you!