1 / 14

2.5 CONTINUITY

2.5 CONTINUITY. Intuitively, the graph of a function can be described as a “continuous curve” if it has not breaks or holes. The graph of a function has a break or hole if any of the following conditions occur:. The function f is undefined at c

windle
Download Presentation

2.5 CONTINUITY

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 2.5 CONTINUITY Intuitively, the graph of a function can be described as a “continuous curve” if it has not breaks or holes. The graph of a function has a break or hole if any of the following conditions occur: • The function f is undefined at c • The limit of f(x) does not exist as x approaches c • The value of the function and the value of the limit at c are different.

  2. Continuous Function

  3. Example: Determine whether the following functions are continuous at x=-3. • Solution: • Observe that • f(x) is not continuous at x=-3 since it’s undefined at x=-3, • g(x) is not continuous at x=-3 since • h(x) is continuous

  4. Continuity on an interval If a function f is continuous at each number in an open interval (a, b), then we say that f is continuous on (a, b). In the case where f is continuous on , we say f is continuous everywhere. A function is continuous from the left at c if A function is continuous from the right at c if

  5. Continuous on a closed interval • 2.5.2 Definition. • A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: • f is continuous on (a, b) • f is continuous from the right at a • f is continuous from the left at b.

  6. Some properties of continuous functions

  7. Continuity of polynomials and rational functions. • 2.5.4 Theorem • A polynomial is continuous everywhere. • A rational function is continuous at every point where the denominator is nonzero, and has discontinuities at the points where the denominator is zero.

  8. Ex: For what values of x is there a discontinuity in the graph of Solution: The function is a rational function, and hence is continuous at every number where there denominator is nonzero. Solve the equation Yields discontinuities at x=2 and x=3.

  9. For example. Find a value of the constant k, if possible, that will make the function continuous everywhere. Solution: since 7x-2 and kx2 are both polynomials, f is continuous for x<1 and x>1. x=1 is the only possible discontinuity for f(x). So if k=5, then f is continuous for all x.

  10. Continuity of compositions For example,

  11. 2.5.6 Theorem • If the function g is continuous at c, and the function f is continuous at g(c), then the composition is continuous at c. • (b) If the function g is continuous everywhere and function f is continuous everywhere, then the composition is continuous everywhere. The absolute value of a continuous function is continuous.

  12. The intermediate-value theorem

  13. Approximating roots using the intermediate-value theorem 2.5.8 Theorem. If f is continuous on [a, b], and if f(a) and f(b) are nonzero and have opposite signs, then there is at least one solution of the equation f(x)=0 in the interval (a, b).

More Related