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**1.4 Continuity, One-Sided Limits, and Intermediate Value Theorem**

Grand Canyon, Arizona Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

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**This function has discontinuities at x=1 and x=2.**

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. 1 2 3 4 This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

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**Removable Discontinuities:**

(You can fill the hole.) Essential Discontinuities: infinite oscillating jump

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**Removing a discontinuity:**

has a discontinuity at Write an extended function that is continuous at Note: There is another discontinuity at that can not be removed.

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**Removing a discontinuity:**

Note: There is another discontinuity at that can not be removed.

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Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. Also: Composites of continuous functions are continuous. examples:

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**THIS IS THE DEFINITION OF CONTINUITY**

A function f is continuous at a point x = c if f (c) is defined 1. 2. 3. THIS IS THE DEFINITION OF CONTINUITY

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**Example f (x) = x – 1 at x = 2. f (2) = a. 1 b. The limit exist! c.**

Therefore the function is continuous at x = 2.

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**Example f (x) = (x2 – 9)/(x + 3) at x = -3 a. - 6 b. c. f (-3) = 0/0**

Is undefined! b. - 6 -3 The limit exists! c. -6 Therefore the function is not continuous at x = -3. You can use table on your calculator to verify this.

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**Continuity Properties**

If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval except for values of x that make the denominator 0. Every polynomial function is continuous. Every rational function is continuous except where the denominator is zero.

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Continuity Summary. Graph on your calculator with a standard window. Functions have three types of discontinuity. Consider - 1. Discontinuity at vertical asymptote. 2. Discontinuity at hole. 3. We have discontinuity with some functions that have a gap.

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**Intermediate Value Theorem**

If a function is continuous between a and b, then it takes on every value between and Because the function is continuous, it must take on every y value between and

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**Intermediate Value Theorem: Intuition**

Traveling on France’s TGV trains, you reach speed of 280 mi/hr. How do you know at some point of train ride you were traveling 100 mi/hr? To go from 0 to 280, must have passed through 100 mi/hr since speed of train changed continuously

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**Intermediate Value Theorem**

Suppose that f is continuous on the closed interval [a,b]. If L is any real number between f(a) and f(b) then there must be at least one number c on the open interval (a,b) such that f(c) = L.

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Limitations of IVT If d(0) = 100 and d(10) = 35, where t is measured in seconds. d is a continuous function, the IVT tells you that at some point between t=0 and t =10, the decibel level reached every value between 35 and 100. It does NOT say anything about: When or how many times (other than at least once) a particular decibel was attained. Whether or not decibel levels bigger than 100 or less than 35 were reached.

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**The Difference Between VROOOOOOOOM and VROOOOOOOM.**

These graphs of PC's noise illustrate that very different behaviors are consistent with the hypothesis that d(t) is continuous and that its values at t=0 and t=10 are 100 and 35 respectively.

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Example 1: Sketch a graph to decide if the cosecant function, f(x) = csc (x) is continuous over the domain [-π, π].

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Example 2 Consider the equation sin x = x – 2 . Use the intermediate Value Theorem to explain why there must be a solution between π/2 and π.

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**Example 3 Consider the function , Calculate f(6), f(-5.5), f(0)**

Can you conclude that there must be a zero between f(6) and f(-5.5)?

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**Is any real number exactly one less than its cube?**

Example 5: Is any real number exactly one less than its cube? (Note that this doesn’t ask what the number is, only if it exists.) Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. Use your calculator to find an approximate solution. F2 1: solve

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**Graphing calculators can sometimes make non-continuous functions appear continuous.**

CATALOG F floor( This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition. Note resolution. The calculator “connects the dots” which covers up the discontinuities.

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**Graphing calculators can make non-continuous functions appear continuous.**

CATALOG F floor( If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph. The open and closed circles do not show, but we can see the discontinuities. GRAPH p

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