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DEFINITION Continuity at a Point f ( x ) is defined on an open interval containing

Three conditions must hold:. DEFINITION Continuity at a Point f ( x ) is defined on an open interval containing x = c . If , then f is continuous at x = c.

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DEFINITION Continuity at a Point f ( x ) is defined on an open interval containing

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  1. Three conditions must hold: DEFINITION Continuity at a Point f (x) is defined on an open interval containing x = c. If , then f is continuous at x = c . If the limit does not exist, or if it exists but is not equal to f (c), we say that f has a discontinuity at x = c.

  2. A function f (x) may be continuous at some points and discontinuous at others. If f (x) is continuous at all points in an interval I, then f (x) is said to be continuous on I. If I is an interval [a, b] or [a, b) that includes a as a left endpoint, we require that Similarly, we require that if I includes b as a right endpoint. If f (x) is continuous at all points in its domain, then f (x) is simply called continuous.

  3. exists but is not equal to f (c) F has a removable discontinuity at x = c.

  4. A “worse” type of discontinuity is a jump discontinuity, which occurs if the one-sided limits and exist but are not equal. Below are two functions with jump discontinuities at c = 2. Unlike the removable case, we cannot make f (x) continuous by redefining f (c). DEFINITION One-Sided Continuity A function f (x) is called: • Left-continuous at x = c if • Right-continuous at x = c if

  5. Piecewise-Defined Function Discuss the continuity of At x = 1, the one-sided limits exist but are not equal: At x = 3, the left- and right-hand limits exist and both are equal to F (3), so F (x) is continuous at x = 3:

  6. f (x) has an infinite discontinuity at x = c if one or both of the one-sided limits is infinite. Notice that x = 2 does not belong to the domain of the function in cases (A) and (B).

  7. Some functions have more “severe” types of discontinuity. For example, oscillates infinitely often between +1 and −1 as x → 0. Neither the left- nor the right-hand limit exists at x = 0, so this discontinuity is not a jump discontinuity.

  8. . It is easy to evaluate a limit when the function in question is known to be continuous.

  9. . Building Continuous Functions THEOREM 1 Basic Laws of Continuity If f (x) and g (x) are continuous at x = c, then the following functions are also continuous at x = c: • (i) f (x) + g (x) and f (x) – g (x) • (ii) kf(x) for any constant k • (iii) f (x) g (x) (iv) f (x)/g (x) if g (c) 0

  10. . Building Continuous Functions THEOREM 2 Continuity of Polynomial and Rational Functions Let P(x) and Q(x) be polynomials. Then: • (i) P(x) is continuous on the real line. • (ii) P(x)/Q(x) is continuous on its domain.

  11. Building Continuous Functions THEOREM 3 Continuity of Some Basic Functions • (i) is continuous on its domain for n a natural #. • (ii) are continuous on the real line. • (iii) is continuous on the real line (for ). • (iv) is continuous for (for ).

  12. . As the graphs suggest, these functions are continuous on their domains. (i.e.) there are no jump discontinuities Because sin x and cosx are continuous, Continuity Law (iv) for Quotients implies that the other standard trigonometric functions are continuous on their domains.

  13. . As the graphs suggest, these functions are continuous on their domains. Because sin x and cosx are continuous, Continuity Law (iv) for Quotients implies that the other standard trigonometric functions are continuous on their domains.

  14. Building Continuous Functions THEOREM 4 Continuity of Composite Functions

  15. Building Continuous Functions THEOREM 4 Continuity of Composite Functions

  16. Building Continuous Functions THEOREM 4 Continuity of Composite Functions

  17. Building Continuous Functions THEOREM 4 Continuity of Composite Functions

  18. The Greatest Integer Function

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