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Limits and Continuity

Lesson objectives Objectives (cont)

Topic 1.11: Defining Continuity at a Point

Topic 1.10: Exploring Types of Discontinuities

Topic 1.12: Confirming Continuity over an IntervalTopic 1.13: Removing Discontinuities

LIM­2: Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

LIM­2.A: Justify conclusions about continuity at a point using the definition.

LIM­2.A.1: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

LIM­2.A.2: A function is continuous at   x=c  provided that f(c) exists, the limit at x=c exists and the function value

is equal to the limit.

LIM­2.B: Determine intervals over which a function is continuous.

Objectives (cont)Lesson objectives

LIM­2.C: Determine values of  x or solve for parameters that make discontinuous functions continuous, if possible.LIM­2.C.1: If the limit of a function exists at a discontinuity in its graph, then it is possible to remove the discontinuity by defining or redefining the value of the function at that point, so it equals the value of the limit of the function as x approaches that point.

LIM­2.C.2: In order for a piecewise­defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on one side of the boundary must equal the value of the expression defining the other side of the boundary, as well as the value of the function at the boundary.

LIM­2.B.1: A function is continuous on an interval if the function is continuous at each point in the interval.LIM­2.B.2: Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.

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Lesson 6: Limits and  Continuity

Now that we have an understanding of limits and using limit notation, we can use limits to define continuity at a point.  This will be a very important concept throughout our course. 

Topic 1.11: Defining Continuity at a Point

EX #1:  A Discovery Exploration.Use the graph below to complete the table.  You should look for three conditions that are necessary to satisfy the definition of continuity. That is, what three conditions must exist in order for y = f(x)  to be continuous at a point x = c ? 

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Analyze this table and write the three conditions for continuity on the first slide. Memorize the rules!

While limits told us where a function intended to go.

Continuity guarantees that the function actually made it there.

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Classifying Discontinuities

Removable or Point (Holes) Non­Removable 

        Jump                       Infinite

                                     1­sided limits exists 

At least oneof the 1­sided 

limits doesn’t exist.

2­sided limit exists

Topic 1.10: Exploring Types of Discontinuities

EX #2:   Find the points (intervals) at which the function is continuous,     and the points at which the function is discontinuous on the                                       

interval 

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Definition of Continuity – More Facts and Theorems

One‑Sided Continuity

A function f (x)  is called

Left‑continuous at x = c  if 

            Right‑continuous at x = c if 

Continuity at a Point

Suppose f (x)  is defined on an open interval containing x = c.Then f is continuous at x = c if 

 

Continuity on an Open Interval: 

A function is continuous on an open interval (a , b ) if it is continuous at each point in the interval.

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 ) and the function is continuous from the 

right at a and continuous from the left at b.

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EX #3:   For                              and            , find f(c) ,                    ,                 and                    . Justify your findings using the three‑ part 

    definition of continuity.  

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Topic 1.12: Confirming Continuity over an Interval

You must be able to confirm continuity without a graph or a calculator by using your knowledge of function behavior for parent functions and their transformations.

EX #4:  Determine whether each function is continuous or not.  If not continuous, justify your answer by using the definition of continuity.

A. B.

Topic 1.13: Removing Discontinuities

EX #4:  Use the definition of continuity to find the value of k so                that the function is continuous for all real numbers.

A. B.

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EX #5:  Given                                                        for what values of x is h(x) 

not continuous?  Justify.

EX #6:  Use the three‑part definition of continuity to create     a system of equations.  Then,  find the values of a     and b so that f (x)  is continuous for all real numbers.

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You know that the function value and the limit value can exist independently of each other.  Let’s summarize the big ideas of continuity on intervals, at points, and one‑sided limits.

EX #7:  Given                                        find each of the following. 

A. B.

C. D.

E.  Is g(x) continuous at x = 3? Justify.

Summary:

If a function is continuous at a point, then the function value and the limit value are the same 

at that point!