Let's first start off with the definition of a continuous function, then we will proceed through some problems that are designed to assist in the development of a greater conceptual understanding of the concept of a continuous function.

A function *f* is continuous at *x=a* provided all
three of the following are truc:

In other words, a function *f* is continuous at a point
*x=a*, when (i) the function *f* is defined at *a*,
(ii) the limit of *f* as *x* approaches *a* from
the right-hand and left-hand limits exist and are equal, and (iii)
the limit of *f* as *x* approaches *a* is equal
to *f(a)*.

Before we go further, let's begin by constructing functions
that are **not** continuous. For each of the following, sketch
a graph of a function that is **not** continuous at *x=2*
when:

a) (i) is true, but (ii) is false.

b) (ii) is true, but (i) is false.

c) (i) and (ii) are true, but (iii) is false.

Now, let's us consider the graph of a function *f* below.

For what numbers *x* in [-3, 3] is *f* not continuous
at *x*?

Next, let's try our hand at another function. For some non-zero
number *a*, define

a) Is

fdefined ata?b) Does exist?

c) Is

fcontinuous ata?

Now, for a bit more complicated problems. If , which of the following statements, if any, must be true?

a)

fis defined ata.b)

f(a)=Lc)

fis continuous ata.

Consider the next function,

Now, (1) find all values of *a* and *b* such that
*f* is continuous at *x=1* and (2) draw the graph of
*f* when *a=1* and *b=-1*.

As we begin to develop a better understanding of continuity, determine whether the following statements are always true or sometimes false. Be sure to justify your answers with an example.

a) If

f(1)<0andf(2)>0, then there must be a pointpin (1, 2) such thatf(p)=0.b) If

fis continuous on [1, 2],f(1)<0andf(2)>0, then there must be a pointpin (1, 2) such thatf(p)=0.c) If

fis continuous on [1, 2] and there is a pointpin (1, 2) such thatf(p)=0, thenf(1)andf(2)must have different signs.d) If

fhas no zeros and is continuous on [1, 2], thenf(1)andf(2)have the same sign.

And, for those of you who love traveling problems: John takes a trip from Portland, OR to Seattle, WA. He leaves at 8:00 am on Monday and arrives at 12:00 pm that same day. He returns on Tuesday, leaving Seattle at 8:00 am and arriving back in Portland at 12:00 pm, retracing exactly the same route. Show that there is a point on the road through with he passes at the same time both days.

This activity was adapted from a Math Excel course taught at Portland State University, spring term 2004 by Joe Ediger and Erin Horst.

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