MATH 8, SECTION 1, WEEK 4  RECITATION NOTES
TA: PADRAIC BARTLETT
Abstract.
These are the notes from Wednesday, Oct. 20th’s lecture, where
we studied several diﬀerent kinds of discontinuous functions.
1.
Random Question
Question 1.1.
Can you ﬁnd a function
f
:
R
→
R
that’s
•
continuous at every rational point
q
∈
Q
, but
•
discontinuous at every irrational point
a
∈
R
\
Q
?
2.
Discontinuity Proofs: A Lemma and a Blueprint
How do we show a function is discontinuous? Speciﬁcally: in our last class, we
described a “blueprint” for showing that a given function was continuous at a point.
Can we do the same for the concept of discontinuity?
As it turns out, we can! Speciﬁcally, we have the following remarkably useful
lemma, proved in Dr. Ramakrishnan’s class on Wednesday:
Lemma 2.1.
For any function
f
:
X
→
Y
, we know that
lim
x
→
a
f
(
x
)
6
=
L
iﬀ there
is some sequence
{
a
n
}
∞
n
=1
with the following properties:
•
lim
n
→∞
a
n
=
L
, and
•
lim
n
→∞
f
(
a
n
)
6
=
L
, and
This lemma makes proving that a function
f
is discontinuous at some point
a
remarkably easy: all we have to do is ﬁnd a sequence
{
a
n
}
∞
n
=1
that converges to
a
on which the values
f
(
a
n
) fail to converge to
f
(
a
). Basically, it allows us to work
in the world of sequences instead of that of continuity; a change that makes a lot
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 Fall '09
 Math, Calculus, TA, Continuous function, Irrational number

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