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Continuous partial derivatives at $x_0$ and differentiability around $x_0$
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Consider function $f: mathbbR^n to mathbbR$. Suppose
1) At $bf x_0$, $f(bf x)$ has partial derivatives $bf J(bf x_0)$.
2) $bf J(bf x)$ is continuous at $bf x_0$.
I know that $f(bf x)$ is differentiable at $bf x_0$.
Question: Is there a neighborhood of $bf x_0$ on which $f(bf x)$ is differentiable?
I know $f(bf x)$ has partial derivatives on a neighborhood of $bf x_0$, however this does not necessarily imply that $f(bf x)$ is differentiable there.
derivatives continuity partial-derivative
asked Jul 15 at 21:17
Guillaume F.
351211
add a comment |Â
up vote
1
down vote
favorite
Consider function $f: mathbbR^n to mathbbR$. Suppose
1) At $bf x_0$, $f(bf x)$ has partial derivatives $bf J(bf x_0)$.
2) $bf J(bf x)$ is continuous at $bf x_0$.
I know that $f(bf x)$ is differentiable at $bf x_0$.
Question: Is there a neighborhood of $bf x_0$ on which $f(bf x)$ is differentiable?
I know $f(bf x)$ has partial derivatives on a neighborhood of $bf x_0$, however this does not necessarily imply that $f(bf x)$ is differentiable there.
derivatives continuity partial-derivative
asked Jul 15 at 21:17
Guillaume F.
351211
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider function $f: mathbbR^n to mathbbR$. Suppose
1) At $bf x_0$, $f(bf x)$ has partial derivatives $bf J(bf x_0)$.
2) $bf J(bf x)$ is continuous at $bf x_0$.
I know that $f(bf x)$ is differentiable at $bf x_0$.
Question: Is there a neighborhood of $bf x_0$ on which $f(bf x)$ is differentiable?
I know $f(bf x)$ has partial derivatives on a neighborhood of $bf x_0$, however this does not necessarily imply that $f(bf x)$ is differentiable there.
derivatives continuity partial-derivative
asked Jul 15 at 21:17
Guillaume F.
351211
Consider function $f: mathbbR^n to mathbbR$. Suppose
1) At $bf x_0$, $f(bf x)$ has partial derivatives $bf J(bf x_0)$.
2) $bf J(bf x)$ is continuous at $bf x_0$.
I know that $f(bf x)$ is differentiable at $bf x_0$.
Question: Is there a neighborhood of $bf x_0$ on which $f(bf x)$ is differentiable?
I know $f(bf x)$ has partial derivatives on a neighborhood of $bf x_0$, however this does not necessarily imply that $f(bf x)$ is differentiable there.
derivatives continuity partial-derivative
asked Jul 15 at 21:17
Guillaume F.
351211
edited Jul 16 at 2:13
asked Jul 15 at 21:17
Guillaume F.
351211
asked Jul 15 at 21:17
Guillaume F.
351211
asked Jul 15 at 21:17
Guillaume F.
351211
351211
add a comment |Â
add a comment |Â
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