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Continuity in R^n
Continuity in R^n
3 questions
Difficulty 2-4
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Foundation
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foundation (2/10)
conceptual
A function
f
:
R
n
→
R
m
is continuous at
x
0
if for every
ϵ
>
0
, there exists
δ
>
0
such that
∥
x
−
x
0
∥
<
δ
implies
∥
f
(
x
)
−
f
(
x
0
)
∥
<
ϵ
. What is the intuition?
Hide and think first
A.
The function is bijective at
x
0
, so it has a unique inverse in a neighborhood of
f
(
x
0
)
B.
Small input changes produce small output changes, with "small enough" quantified by
δ
controlling
ϵ
C.
The function preserves distances, so
∥
f
(
x
)
−
f
(
y
)
∥
=
∥
x
−
y
∥
for all
x
,
y
D.
The function is differentiable at
x
0
, so its derivative is well-defined with no degenerate cases
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