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 Dx being small (
maybe even 0), let
Df º f(a + Dx) - f(a).
To emphasize the dependence of Df on a and Dx, we somtimes write
Df as Df(a) or Df(a,Dx).
Then our working definitions of continuity at a become
·
(Working JL). The function f
is continuous at a, if we can make Df(a) is assmallasdesired as
by requiring that Dx be closeenoughto0.
·
(More Informal) The function f is continuous at a,
if Df(a) is small whenever Dx is small enough.
e- d Definition. The function f is continuous at a means: For every e > 0, there is some d > 0 such that, for all x, if |x -a| < d, then |f(x) - f(a)| < e.
Variation e- d Definition. The function f is continuous at a means: For every e > 0, there is some d > 0 such that, if |Dx| < d, then |Df(a,Dx| < e.
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) ¹ 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: (e-d). We must show that:
For every e > 0, there is some
d > 0 such that, for all x, if |x -a| < d,
then |f(g(x)) - f(g((a))| < e.
Fix e > 0. Use the continuity of f at g(a) to find a
d1 > 0 such that, for all y, if |y - g(a)| < d1,
then |f(y) - f(g(a))| < e.
Now use the continuity of
g at a to to find a
d2 > 0 such that, for all x, if |x - a| < d2,
then |g(x) - g(a)| < d1.
Then for all x, if |x - a| < d2,
then |g(x) - g(a)| < d1 and |f(g(x)) - f(g((a))| < e.
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 Î (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 Î (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
¹ a " but could have included it without changing the
meaning.
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