Pg 43, Example 3.1 – For each number x, define . Then the function
is continuous. To verify this, we select a point
x
0
in R, and we will show that the function is continuous at x
0
. Let
be a sequence that converges to x
0
. By the sum
and product properties of convergent sequences,
Thus,
is continuous at x
0
.
Pg 43, Example 3.2 – Define for . Then the function
is continuous. To verify this, we select a nonnegative number
and let
be a sequence of nonnegative numbers that converges to . But then the sequence
converges to ; that is,
Thus,
is continuous at x
0
.
Pg 44, Example 3.3 – Define the function by
Prove that there is no point in R (domain) in which the function f is continuous.
Pf:
Let x
0
be in R (domain). For each natural number n, by Thm 1.11, we may choose a rational number, which
we will label , in the interval
and an irrational, which we will label , in the interval . But for each natural number n,
and , so
Since both of the sequences
and
converges to x
0
, it is not possible for
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 Spring '11
 Higdon
 Continuous function, pg, convergent sequences

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