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that the mathematical definition of continuity corresponds closely
with the meaning of the word continuity in everyday language. (A
continuous process is one that takes place gradually, without interruption
or abrupt change.)
Definition 13
A function f(x) is continuous at a number a if
lim
x→a
f(x)=f(a).
If f(x) is not continuous at a,wesayf(x) is discontinuous at a,or
f(x) has discontinuity at a. Notice that Definition implicitly requires
three things if f(x) is continuous at a:
1. f(a) is defined (that is, a is in the domain of f(x)).
2. lim
x→a
f(x) exists (so f(x) mustbedefinedonanopeninterval
that contains a).
3. lim
x→a
f(x)=f(a).
The definition says that f(x) is continuous at x
0
if f(x) approaches
f(x
0
) as x approaches x
0
.
Some texts introduce the increment of x as x−x
0
. The corresponding
change in y = f(x) is ∆y = f(x) − f(x
0
).
Assume that y = f(x) is defined in some open interval containing
x
0
. Clearly, it is continuous at x
0
if lim
∆x→0
∆y =0or, which is the
same,
lim
∆x→0
[f(x
0
+∆x) − f(x
0
)] = 0.
We can distinguish between different kinds of discontinuity at x
0
.
Example 7
Give an example (graphically, or by a formula, or both) of a
function which
12
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