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and f(x) is continuous from the left at x = a number if
lim
x→a−
f(x)=f(a), or f(a − 0) = f(a)
(as you have noticed earlier, the Western notation for one-sided limits
differs slightly from the Russian one).
Definition 15
A function f(x) is continuous on an interval if it is continuous
at every number in the interval. (At an endpoint of the interval
we understand continuous to mean continuous from the right or
continuous from the left.
Example 8
1. A function f(x) = arctan
1
x
is not defined at 0. We conclude
that it is discontinuous at 0. We cannot remove this discontinuity
since
lim
x→0−
arctan
1
x
= −
π
2
, lim
x→0+
arctan
1
x
=
π
2
.
2. A function y =
sin x
x
is not defined at 0. This is a removable
discontinuity since
f(−0) = f(+0) = lim
x→0
sin x
x
=1.
3. A function f(x)=
1
x − 1
has a jump discontinuity at x =1.
Clearly, y>0 if x>1 and y<0 if x<1 from where we
conclude about one-sided limits f(1 + 0) = +∞, f(1 − 0) =
−∞, and, finally, about the jump discontinuity at x =1(which
cannot be removed).
4. A function f(x)=sin
1
x
has one other type of discontinuity at
0.
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