21.F. Let f be defined on R to R by f(x) = x + 2x? sin (1/x), x * 0, = 0, x = 0. Then Df(0) is one-one but ƒ has no inverse near x = 0.

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Chapter2: Second-order Linear Odes
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please help with exercise 21.F

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21.D. Let f be the mapping in the preceding exercise. Show that f is locally
one-one at every point except 0
21.E. Let f be in Class C' on R" onto Rº and suppose that f has an inverse. Is
it true that for each x in RP, then Df(x) is a one-one linear function which maps
R? onto Re?
21.F. Let f be defined on R to R by
(0, 0), but f is not one-one on R?.
f(x)
= x + 2x? sin (1/r),
* * 0,
= 0,
* = 0,
Then Df(0) is one-one but f has no inverse near x = 0.
21.G. Let f be a function on R to R? which is differentiable on a neighborhood
of a point c and such that Df(c) has an inverse. Then is it true that f has an
inverse on a neighborhood of c?
Transcribed Image Text:21.D. Let f be the mapping in the preceding exercise. Show that f is locally one-one at every point except 0 21.E. Let f be in Class C' on R" onto Rº and suppose that f has an inverse. Is it true that for each x in RP, then Df(x) is a one-one linear function which maps R? onto Re? 21.F. Let f be defined on R to R by (0, 0), but f is not one-one on R?. f(x) = x + 2x? sin (1/r), * * 0, = 0, * = 0, Then Df(0) is one-one but f has no inverse near x = 0. 21.G. Let f be a function on R to R? which is differentiable on a neighborhood of a point c and such that Df(c) has an inverse. Then is it true that f has an inverse on a neighborhood of c?
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