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Understanding Continuity and Discontinuity in Functions

This guide explores the concepts of continuity and discontinuity in mathematical functions. We will outline the criteria for a function to be continuous at a specific point and identify different types of discontinuities such as removable, jump, and infinite discontinuities. The Intermediate Value Theorem (IVT) will also be discussed, showing how continuous functions on a closed interval take on every value between their endpoints. Examples will illustrate how to determine continuity across specific points and intervals, enabling a deeper understanding of function behavior.

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Understanding Continuity and Discontinuity in Functions

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  1. 3.2: Continuity Objectives: To determine whether a function is continuous Determine points of discontinuity Determine types of discontinuity Apply the Intermediate Value Theorem

  2. CONTINUITY AT A POINT—No holes, jumps or gaps!! A function is continuous at a point c if: 1.) f(c) is defined 2.) exists 3.) = f(c) A FUNCTION NEED NOT BE CONTINUOUS OVER ALL REALS TO BE A CONTINUOUS FUNCTION

  3. Does f(2) exist? Does exist? Does exist? Is f(x)continuous at x = 2? Do the same for x= 1, 3, and 4.

  4. Removable discontinuity • Limit exists at c but f(c)≠ the limit • Can be fixed. Set f(c) = • This is called a continuous extension

  5. Example: Find the values of x where the function is discontinuous.

  6. OTHER TYPES OF DISCONTINUITY • JUMP: ( RHL ≠ LHL) • INFINITE: • OSCILLATING

  7. Continuity on a closed interval A function is continuous on a closed interval [a,b] if: • It is continuous on the open interval (a,b) • It is continuous from the right at x=a: • It is continuous from the left at x=b:

  8. example IT IS CONTINUOUS ON ITS DOMAIN.

  9. Key Functions and where they are continuous

  10. A continuous function is one that is continuous at every point in its domain. It need not be continuous on all reals. Where are the functions discontinuous? If it is removable discontinuity, fix it!!

  11. Where are the functions discontinuous? Fix removable!

  12. Find the value of the constant k that makes the function continuous.

  13. Intermediate Value Theorem (IVT) A function y=f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). THE FUNCTION MUST BE CONTINUOUS ON INTERVAL!!

  14. If f is continuous on [a,b] and f(A)and f(b) differ in signs, then there must be zero on [a,b]. 1. Show that f(x)=x3+2x-1 has a zero in [0, 1]. 2. Is any real number exactly 1 less than its cube?

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