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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

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2.5 CONTINUITY

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  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).

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