Show that f is continuous on (−∞, ∞).
f(x) =
 |
| 1 − x2 |
if x ≤ 1 |
| ln(x) | if x > 1 |
On the interval
(−∞, 1),
f is function; therefore f is continuous
on
(−∞, 1).
On the interval
(1, ∞),
f is function; therefore f is continuous
on
(1, ∞).
At
x = 1,
lim
x→1−f(x)
= lim x→1−
 |
 |
=
,
and
lim
x→1+f(x)
= lim x→1+
|
|
=
,
so
lim x→1
f(x) =
.
Also,
f(1) =
.
Thus, f is continuous at
x = 1.
We conclude that f is continuous on
(−∞, ∞).
Answer
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