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(Section 2.8: Continuity)
2.8.20
FOOTNOTES
1.
A function with domain
±
that is only continuous at 0.
(Revisiting Footnote 1 in Section
2.1.) Consider the following function
f
.
fx
()
=
0,
if
x
is a rational value
x
,i
f
x
is an irrational value
±
²
³
f
is continuous at 0, because
f
0
=
0 , and we can use the Squeeze (Sandwich) Theorem to
prove that
lim
x
±
0
=
0 , also. The discontinuities at the nonzero real numbers are neither
removable, jump, nor infinite.
2.
Infinite discontinuities: alternate definition.
Some textbooks only require that
±²
or
³²
as
x
±
a
+
or
x
±
a
²
. Such a definition would allow us to say that
the following function has an infinite discontinuity at 0.
=
1
x
f
x
is a rational value
±
1
x
f
x
is an irrational value
²
³
´
´
µ
´
´
If we adopt the onepoint compactification of the real numbers, also known as the real
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This note was uploaded on 12/29/2011 for the course MATH 150 taught by Professor Sturst during the Fall '10 term at SUNY Stony Brook.
 Fall '10
 sturst
 Continuity

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