Pay attention to the domain of the function. In particular, if
f is continuous at a, then a and all points sufficiently
close to a are in the domain of f.
Working Definition. The function f is continuous at a, if we can make f(x) as close as we like to f(a) by requiring that x be sufficiently close to a1.
•
(Working JL) The function f
is continuous at a, if f(x) = f(a) + assmallasdesired whenever x = a +closeenoughto0.
•
(More Informal) The function f is continuous at a,
if f(x) is close to f(a) whenever x is close enough to a.
Thinking of a as fixed, and letting ∆x being small (
maybe even 0), let
∆f ≡ f(a + ∆x) − f(a).
To emphasize the dependence of ∆f on a and ∆x, we somtimes write
∆f as ∆f(a) or ∆f(a,∆x).
Then our working definitions of continuity at a become
•
(Working JL). The function f
is continuous at a, if we can make ∆f(a) is assmallasdesired as
by requiring that ∆x be closeenoughto0.
•
(More Informal) The function f is continuous at a,
if ∆f(a) is small whenever ∆x is small enough.
ϵ− δ Definition. The function f is continuous at a means: For every ϵ > 0, there is some δ > 0 such that, for all x, if |x −a| < δ, then |f(x) − f(a)| < ϵ.
Variation ϵ− δ Definition. The function f is continuous at a means: For every ϵ > 0, there is some δ > 0 such that, if |∆x| < δ, then |∆f(a,∆x| < ϵ.
By the fundamental limit theorems, If f and g are two
functions continuous at a, then
•
f + g is continuous at a,
•
f ·g is continuous at a,
•
f / g is continuous at a, provided g(a) ≠ 0.
Compositions Theorem 2. If g is continuous at a and f is continuous at g(a), then f °g is continuous at a.
Proof: (ϵ−δ). We must show that:
For every ϵ > 0, there is some
δ > 0 such that, for all x, if |x −a| < δ,
then |f(g(x)) − f(g((a))| < ϵ.
Fix ϵ > 0. Use the continuity of f at g(a) to find a
δ1 > 0 such that, for all y, if |y − g(a)| < δ1,
then |f(y) − f(g(a))| < ϵ.
Now use the continuity of
g at a to to find a
δ2 > 0 such that, for all x, if |x − a| < δ2,
then |g(x) − g(a)| < δ1.
Then for all x, if |x − a| < δ2,
then |g(x) − g(a)| < δ1 and |f(g(x)) − f(g((a))| < ϵ.
Continuity on Intervals
A function f defined on an interval I = (a,b) is
continuous on I if f is continuous at x for very x ∈ (a,b).
Continuity of a function on a non open interval requires a
modification of the definition of continuity at the included
endpoints. A function f defined on a closed interval I = [a,b] is continuous on I if f is continuous at x
for very x ∈ (a,b), right continuous at a -
limx → a+ f(x) = f(a) - and left continuous at b -
limx → b− f(x) = f(b). Make the obvious modifications
if I = [a,b) or I = (a,b]
Footnotes:
1Note that we have omitted the phrase "but
≠ a " but could have included it without changing the
meaning.
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