how do you determine which piecewise functions are continuous and discontinuous ?
Answers
Answer:
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Step-by-step explanation:
In most cases, we should look for a discontinuity at the point where a piecewise defined function changes its formula. You will have to take one-sided limits separately since different formulas will apply depending on from which side you are approaching the point. Here is an example.
Let us examine where
f
has a discontinuity.
f
(
x
)
=
⎧
⎪
⎨
⎪
⎩
x
2
if
x
<
1
x
if
1
≤
x
<
2
2
x
−
1
if
2
≤
x
,
Notice that each piece is a polynomial function, so they are continuous by themselves.
Let us see if
f
has a discontinuity
x
=
1
.
lim
x
→
1
−
f
(
x
)
=
lim
x
→
1
−
x
2
=
(
1
)
2
=
1
lim
x
→
1
+
f
(
x
)
=
lim
x
→
1
+
x
=
1
Since both limits give 1,
lim
x
→
1
f
(
x
)
=
1
f
(
1
)
=
1
Since
lim
x
→
1
f
(
x
)
=
f
(
1
)
, there is no discontinuity at
x
=
1
.
Let us see if
f
has a discontinuity at
x
=
2
.
lim
x
→
2
−
f
(
x
)
=
lim
x
→
2
−
x
=
2
lim
x
→
2
+
f
(
x
)
=
lim
x
→
2
+
(
2
x
−
1
)
=
2
(
2
)
−
1
=
3
Since the limits above are different,
lim
x
→
2
f
(
x
)
does not exist.
Hence, there is a jump discontinuity at
x
=
2
.
I hope that this was helpful.