LC - Continuity and Intermediate Value Theorem Lesson

Continuity and Intermediate Value Theorem

image of pencil Physical phenomena such as the velocity of a car or the rising and setting of the sun vary continuously with time. However, other situations such as electric currents are not continuous, but rather are described as discontinuous. Geometrically, a continuous function over an interval appears as a graph with no break in it. Such a graph can be drawn without picking up your drawing tool from the paper.

Definition of Continuity

A function f is continuous at a number a if equation image indicator $LaTeX: \lim _{x\to a}f\left(x\right)=f\left(a\right)$ $\lim _{x\to a}f\left(x\right)=f\left(a\right)$ . View the presentation below on an informal introduction to continuity and discontinuities.

Continuity Test

A function f is continuous at x = a if and only if all three of the following conditions are true:

f(a) is defined
$LaTeX: \lim _{x\to a}f\left(x\right)$ $\lim _{x\to a}f\left(x\right)$ exists
$LaTeX: \lim _{x\to a}f\left(x\right)=f\left(a\right)$ $\lim _{x\to a}f\left(x\right)=f\left(a\right)$

Types of Discontinuities

image of athlete mid-flight in high jump A function is discontinuous at x = a if the $LaTeX: \lim _{x\to a}f\left(x\right)$ $\lim _{x\to a}f\left(x\right)$ equation image indicator does not exist, or if it exists but does not equal f(a). Discontinuities are of two types: removable and non-removable. In each case the graph cannot be drawn without lifting pencil from paper due to a hole, break, or jump in the graph. A discontinuity at f(a) is removable if the function can be redefined at the number a so as to make the function continuous. Both jump and infinite discontinuities are considered non-removable. A jump discontinuity occurs when the left- and right-sided limits exist at x = a but the limits do not agree. This behavior is exhibited by the graph of the function f(x) jumping at x = a. An infinite discontinuity exists at a specific value of c as it is approached from one or both sides and the function f approaches infinity or negative infinity.

View the presentation below on the Continuity Using Limits.

Properties of Continuous Functions

If f and g are continuous functions at a and c is a constant, then the following functions are also continuous at a:

Sum and difference: $LaTeX: f\pm g$ $f\pm g$
Scalar multiple: cf
Product: fg
Quotient: $LaTeX: \frac{f}{g}\:if\:g\left(x\right)\ne0$ $\frac{f}{g}\:if\:g\left(x\right)\ne0$

It is often expedient to also know that the following types of functions are continuous at every number in their domains: polynomial, rational, radical (root), trigonometric, inverse trigonometric, exponential, and logarithmic.

Continuity of a Composite Function

If g is continuous at a and f is continuous at g(a), then the composite function (fοg)(x) = f(g(x)) is continuous at a.

Intermediate Value Theorem

Did You Know? The intermediate value theorem plays an important role in how graphing devices work. A computer calculates a finite number of points on the graph and turns on the pixels that contain these calculated points. It assumes that the function is continuous and takes on all of the intermediate values between two consecutive points. The Intermediate Value Theorem states that if f is continuous on the closed interval [a, b] and N is any number between f(a) and f(b), where f(a)≠f(b), then there exists a number c in (a, b) such that f(c) = N. A simpler interpretation of what the Intermediate Value Theorem says is that a continuous function will take on all values between f(a) and f(b). One of the most important applications of the Intermediate Value Theorem is locating roots (zeros) of equations.

Consequences Of Continuity
If f is continuous on [a,b] and f(a) and f(b) differ in sign, the Intermediate Value Theorem guarantees the existence of at least one zero of f in the closed interval [a,b]
Consider the following polynomial f(x)=x³-x²+4x-3.
f(x) is continuous for all x values since it is a polynomial.
Since f(0)=-3, and f(1)=1, it follows that f(x)=0 for some x between 0 and 1.

Continuity and Intermediate Value Theorem Practice

Continuity and Intermediate Value Theorem: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of continuity, discontinuity, and the Intermediate Value Theorem.

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