LIMITS AND CONTINUITY (18.014, FALL 2015)
In lecture I defined limits and continuous functions slightly differently from how Apostol
does in the textbook for this class. These notes present the definitions that I gave. In the
final section of these notes, I explain how they differ from the definitions used by Apostol.
1. Limit points
Definition. Let S ⊆ R be a set of real numbers. Then a real number p is a limit point of S
if for any δ > 0, there exists x ∈ S such that
0 < |x − p| < δ.
Note that a limit point of S does not need to belong to S itself, and elements of S do
not need to be limit points of S. An element of S that is not a limit point of S is called an
isolated point of S.
Example 1. The set of integers Z has no limit points. Every element of Z is an isolated
point.
Example 2. An open interval (a, b) has no isolated points. The set of limit points of (a, b) is
the closed interval [a, b].
2. Limits
Definition. Let f : S → R be a real-valued function defined on some set of real numbers
S ⊆ R. Let p be a limit point of S. If b ∈ R satisfies the condition:
for any ε > 0, there exists δ > 0 such that
|f (x) − b| < ε whenever 0 < |x − p| < δ,
then we say that the limit of f (x) as x approaches p is b and we write
lim f (x) = b.
x→p
Note that the limit is unique if it exists; this follows easily from the triangle inequality
and the definition of a limit point.
Example 3. If f is a constant function with value c, then
lim f (x) = c
x→p
at any limit point p.
Example 4. If f is the identify function f (x) = x, then
lim f (x) = p
x→p
at any limit point p, since we may take δ = ε in the definition of the limit.
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Example 5 (jump discontinuity). Let f : R → R be defined by
(
1 if x ≤ 0
f (x) =
.
2 if x > 0
Then limx→0 f (x) does not exist.
3. One-sided limits
In general limits behave nicely with respect to restrictions of functions.
Proposition 3.1. Let T ⊆ S ⊆ R be nested subsets of the reals. Let f : S → R be a function
and let p be a limit point of T . If limx→p f (x) exists, then
lim (f |T )(x) = lim f (x),
x→p
x→p
where f |T : T → R is the restriction of f to T .
Definition (one-sided limits). Let S ⊆ R and f : S → R. Let p be a limit point of S. Let
S>p = {x ∈ S | x > p} and S<p = {x ∈ S | x < p}
be subsets of S. Then if p is a limit point of S>p , define the limit of f (x) as x approaches p
from above to be
lim+ f (x) := lim (f |S>p )(x).
x→p
x→p
Similarly define
lim f (x) := lim (f |S<p )(x).
x→p−
x→p
Proposition 3.1 tells us that if a regular limit exists, then it is equal to the one-sided limits
at that point. (It is also possible to show that if the one-sided limits at a point exist and are
equal, then the regular limit exists.) This gives a simple argument for the nonexistence of
the limit in Example 5: if the limit existed, then it would be equal to both of its one-sided
limits, but those are nonequal in this case.
4. Continuous functions
Definition. Let S ⊆ R and f : S → R. Let p be an element of S. Then f is continuous at
p if either
(a) p is an isolated point of S, or
(b) limx→p f (x) = f (p).
We say f is continuous if it is continuous at every p ∈ S.
Example 6. Every function on Z is continuous, because every point is isolated.
Example 7. The constant and identity functions are continuous.
Example 8. The “jump” function of Example 5 is continuous everywhere except at x = 0.
Example 9. By Proposition 3.1, if f is continuous at p then any restriction of f to a set
containing p will still be continuous at p. This means that the restriction of any function to
the set of points at which it is continuous will be a continuous function.
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5. Comparison with Apostol
The main difference between our treatment of limits and continuity and that of Apostol is
that he only defines the limit of f (x) as x approaches p if the domain of f contains an open
interval containing p (with the exception of the point p itself, where f does not have to be
defined). This has the advantage that Apostol does not need to introduce the concept of a
limit point (p is automatically a limit point in this setting), but it applies to fewer functions.
Our definition agrees with the more general setting of topological continuity.
The main advantage of our definition of the limit in this course is that we don’t need to
talk about one-sided limits when discussing functions defined on a closed interval [a, b]. For
example, Apostol needs to use one-sided limits to define what it means for a function to
be continuous on the endpoints of [a, b] (see the parenthetical comment at the end of the
statement of Theorem 3.4 in Apostol), while we can just use regular limits. In other words,
let f : [a, b] → R be a function on a closed interval. Then we have
lim f (x) = lim+ f (x),
x→a
x→a
while Apostol does not define the first limit in this setting and is forced to use the second.
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