2.3 Continuity and Intermediate Value Theorem

2.3 Continuity and Intermediate Value Theorem

Continuity • A function is continuous if you can draw the graph without picking up your pencil. Definition: A function y = f(x) is continuous at an interior point c in its domain if (the limit has to equal the value of the function at c)

Continuity (at endpoints) • A function y = f(x) is continuous at a left endpoint a or a right endpoint b in its domain if or If a function is not continuous anywhere on its graph, we say there is a discontinuity at that point.

Continuity - Graphically Continuous at a? Yes Continuous at b? Yes b a c Continuous at c? Yes

Discontinuities continuous for all real numbers? • Is This is an example of a nonremovable discontinuity at x = 0. (vertical asymptote)

Discontinuities • Is f(x) = [x] continuous for all real numbers? Since the limit does not exist, we can say the function is discontinuous. This is another example of a nonremovable discontinuity at x = 0. (jump)

Discontinuities • Is the function continuous for all real numbers? Sin constantly oscillates, so we cannot say that these two limits are the same. Thus, the limit as x approaches 0 does not exist and there is a discontinuity here. Another nonremovable discontinuity. (infinite oscillation)

Discontinuities • Is continuous everywhere? (1, 2) because f(1) does not exist. This is an example of a removable discontinuity at x = 1. (hole)

Removing a Discontinuity (hole) • How can we make continuous? The only place in question is at x = 1. When you can cancel out like factors in the top and bottom, this means there is a hole at where the denominator was equal to 0. To be continuous, the limit as x approaches 1 has to be the value of the function at 1. Therefore, to be continuous, f(1) = 2.

Removing a Discontinuity (hole) • How can we make continuous? We can now turn the original function into a piecewise so that f(1) not only exists, but is equal to the limit as x approaches 1, which was 2.

Properties of Continuous Functions • If two functions are continuous, then • The sum of the functions is also continuous. • The difference of the functions is also continuous • The product of the functions is also continuous. • The result of multiplying one of the functions by a constant is also continuous. • The quotient of the functions is also continuous provided the denominator ≠ 0

Theorem Involving Composites • If f(x) = │x │ and g(x) = cos x, what is f(g(x))? f(g(x)) = │cos x│ If f and g are both continuous functions, the composites of two continuous functions is also always continuous.

Big Theorem #1:Intermediate Value Theorem • A function f(x) that is continuous on the interval [a, b] takes on every value between f(a) and f(b). f(b) f(a) a b

Intermediate Value Theorem • Show using the IVT that has a root between x = 2 and x = 3. f(x) is discontinuous only at x = –1, so on the interval [2, 3] the function is continuous. Since f is cont. on [2, 3] and f(2) and f(3) have opposite signs, there is a value c in the Interval where f(c) = 0 by the Intermediate Value Theorem.

2.3 Continuity and Intermediate Value Theorem