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The Banach Fixed-Point Theorem
Recognizing a Contraction
MCQ
Which of the following statements about contraction mappings are true?
Possible answers
If
f
f
f
is differentiable on an interval
I
I
I
with
∣
f
′
(
x
)
∣
≤
k
<
1
\lvert f'(x) \rvert \leq k < 1
∣
f
′
(
x
)∣
≤
k
<
1
, then
f
f
f
is
k
k
k
-contracting on
I
I
I
It is enough that
∣
f
(
x
)
−
f
(
y
)
∣
<
∣
x
−
y
∣
\lvert f(x) - f(y) \rvert < \lvert x - y \rvert
∣
f
(
x
)
−
f
(
y
)∣
<
∣
x
−
y
∣
for all
x
≠
y
x \neq y
x
=
y
for
f
f
f
to be a contraction
The cosine function is a contraction on
[
0
;
1
]
[0 \,;\, 1]
[
0
;
1
]
The function
x
↦
x
+
1
x \mapsto x + 1
x
↦
x
+
1
is a contraction on
R
\mathbb{R}
R
, because it is
1
1
1
-Lipschitz
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