Rigorous Definition of Continuity and Limits of Functions
Leo Livshits
Nov. 6
1 Sequential definition of continuity
In your previous Calculus courses you have discussed continuous functions, but in many
cases no rigorous definition of continuity was presented. We shall remedy this at present.
Let us agree that D will stand for a subset of R, most often an interval of some kind.
1 Definition. Given a function f : D −→ R and a sequence A = (a1 , a2 , a3 , . . .) in D, we
write f (A) for the sequence (f (a1 ), f (a2 ), f (a3 ), . . .).
In the language of compositions of functions f (A) is simply the composition f ◦ A.
Recall that sequence A is a function from N to D, and f is a function from D to R, so that
f ◦ A is a function from N to R. This is a good time to refresh your memory about
compositions of functions!
2 Example. If H is the Harmonic sequence, then sin(H) is the sequence
1
1
(sin(1), sin( ), sin( ), . . .).
2
3
3 Definition. Given a function f : D −→ R, and a number L in D, we say that f is
continuous at L provided the following implication holds true:
If A is a sequence in D and A −→ L, then f (A) −→ f (L).
In common terms, a function is continuous at a point if it preserves convergence of
sequences at that point.
4 Examples.
1. Function f (x) = x is continuous at every real L (in this case we say that f is
continuous on R). Indeed, here f (A) = A and f (L) = L so that the implication in
definition 3 is trivially true.
2. Constant function g(x) = C is continuous on R.
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Limit laws for sequences immediately yield the following continuity laws:
5 Theorem. If functions f : D −→ R and g : D −→ R are both continuous at a given
number M in D, then so are
1. f + g
2. αf (for any α ∈ R)
3. f − g
4. f · g
When g(M ) 6= 0, we also have that
5.
1
g
is continuous at M
6.
f
g
is continuous at M
Proof. Claim 3 follows from claims 11 and 2. Similarly claim 6 follows from claims 4 and
5. (Do you see how?) Let us prove part 4, and you shall see how the rest of the proofs
work. We need to verify the truth of the following implication:
If A is a sequence in D and A −→ M , then (f · g)(A) −→ (f · g)(M ).
(♦)
Since (f · g)(F) = f (F) · g(F) by definition, ♦ can be rewritten as
“If A is a sequence in D and A −→ M , then f (A) · g(A) −→ f (M ) · g(M ).”
Let us start: suppose that A is a sequence in D and A −→ M .
(4)
[We aim to show that f (A) · g(A) −→ f (M ) · g(M ).]
Since f and g are both continuous at M , we can infer from (4) that f (A) −→ f (M ) and
g(A) −→ g(M ). The desired conclusion now follows from the limit rule for products of
sequences (for example see Theorem 4.1.5 in Belding and Mitchell).
6 Corollary.
1. f (x) = xk is continuous on R for very k ∈ N0 .
2. Every polynomial is continuous on R.
3. A quotient of two polynomials is continuous at every point of its domain.
(Proof of claim 1 is by induction via claim 4 of Theorem 5 and Example 4. Proof of
claim 2 uses claim 1 and claims 11 and 2 of Theorem 5 via induction on the degree of the
polynomial. Proof of claim 3 uses claim 6 of Theorem 5 and claim 2.)
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What about continuity and composition? No problem:
7 Theorem. Suppose z is a point in the domain D of the composition f ◦ g, where g is
continuous at z, and f is continuous at g(z); (i.e. g is “nice” at the input it gets when z
is fed to the composition function, and f is “nice” at the input it gets from g). Then f ◦ g
is continuous at z.
Proof. We need to verify the truth of the following implication:
If A is a sequence in D and A −→ z, then (f ◦ g)(A) −→ (f ◦ g)(z).
(♦♦)
Since (f ◦ g)(F) = f (g(F)) by definition, ♦♦ states:
If A is a sequence in D and A −→ z, then f (g(A)) −→ f (g(z)).
Let us prove this implication. Suppose that A is a sequence in D and A −→ z.
(44)
[We shall show that f (g(A)) −→ f (g(z)).]
Since g is continuous at z, we can infer from 44 that g(A) −→ g(z). Since f is continuous
at g(z), we conclude that f (g(A)) −→ f (g(z)) as required.
Finally, you may now assume that root, trigonometric, exponential and logarithmic functions are all continuous at every point of their domain. We may
eventually discuss the reasons why this is true, but this must wait till the time we are able
to think more about rigorous definitions of these functions.
2 Introducing Limits of functions
We can give a precise definition of a function limit as well.
6=
8 Notation. We shall write [un ] −→ α to indicate that the sequence [un ] converges to α
and all terms of [un ] are distinct from α.
9 Definition. A real number α is said to be a limit point of D (which still stands for a
6=
subset of R), if [un ] −→ α for at least one sequence [un ] in D.
10 Example. Let S = (0, 1) ∪ (1, 2] ∪ {3}. Then 0 is a limit point of S, so are 1, 2 and 35 ,
but 3 and 7 are not limit points of S. (The set of limit points of S is exactly [0, 2].)
11 Definition. Suppose f : D −→ R and α is a limit point of D. We say that
lim f (x) = F
x→α
if
6=
[xn ] −→ α
xn ∈ D
)
=⇒ [f (xn )] −→ F.
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12 Remark. Just like in the case of continuity, operations on limits of functions follow the
same basic rules as operations on limits of sequences, for obvious reasons. These rules can
be found in section 2.3 of Belding and Mitchell. Please read the rules themselves, but skip
the proofs, since B&M follow a different approach in their development of limit theory.
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