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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Chapter2: Second-order Linear Odes
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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?
21.H. Let f be a funetion on Rº to RP. If f is differentiable at c and has a
differentiable inverse, then is it true that Df(c) is one-one?
21.1. Suppose that f is differentiable on a neighborhood of a point c and that
if e > 0 then there exists 6(e) > 0 such that if |r - c| < ô(e), then |Df(x) (2) –
Df(c) (2) < elz| for all z in RP. Prove that the partial derivatives of f exist and
are continuous at c.
21.J. Suppose that Lo is a one-one linear function on Rr to R. Show that
there exists a positive number a such that if L is a linear function on Rr to R
satisfying
Transcribed Image Text: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? 21.H. Let f be a funetion on Rº to RP. If f is differentiable at c and has a differentiable inverse, then is it true that Df(c) is one-one? 21.1. Suppose that f is differentiable on a neighborhood of a point c and that if e > 0 then there exists 6(e) > 0 such that if |r - c| < ô(e), then |Df(x) (2) – Df(c) (2) < elz| for all z in RP. Prove that the partial derivatives of f exist and are continuous at c. 21.J. Suppose that Lo is a one-one linear function on Rr to R. Show that there exists a positive number a such that if L is a linear function on Rr to R satisfying
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