Continuity of partial derivative
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I come up with this question while working on an exercise.
Suppose that $f(x,y)$ is defined in some neighborhood of $(x_0,y_0)$, $frac{partial^2 f}{partial x partial y}$ exists and is continuous at $(x_0,y_0)$, $partial f/partial x$ exists. Is it true that $partial f/partial x$ is also continuous at $(x_0,y_0)$? I think of Leibniz rule but its hypothesis include continuity in a neighborhood and not just at a point.
I appreciate any help.
analysis continuity partial-derivative
asked Dec 2 at 4:08
Jiu
3409
add a comment |
up vote
1
down vote
favorite
I come up with this question while working on an exercise.
Suppose that $f(x,y)$ is defined in some neighborhood of $(x_0,y_0)$, $frac{partial^2 f}{partial x partial y}$ exists and is continuous at $(x_0,y_0)$, $partial f/partial x$ exists. Is it true that $partial f/partial x$ is also continuous at $(x_0,y_0)$? I think of Leibniz rule but its hypothesis include continuity in a neighborhood and not just at a point.
I appreciate any help.
analysis continuity partial-derivative
asked Dec 2 at 4:08
Jiu
3409
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I come up with this question while working on an exercise.
Suppose that $f(x,y)$ is defined in some neighborhood of $(x_0,y_0)$, $frac{partial^2 f}{partial x partial y}$ exists and is continuous at $(x_0,y_0)$, $partial f/partial x$ exists. Is it true that $partial f/partial x$ is also continuous at $(x_0,y_0)$? I think of Leibniz rule but its hypothesis include continuity in a neighborhood and not just at a point.
I appreciate any help.
analysis continuity partial-derivative
asked Dec 2 at 4:08
Jiu
3409
I come up with this question while working on an exercise.
Suppose that $f(x,y)$ is defined in some neighborhood of $(x_0,y_0)$, $frac{partial^2 f}{partial x partial y}$ exists and is continuous at $(x_0,y_0)$, $partial f/partial x$ exists. Is it true that $partial f/partial x$ is also continuous at $(x_0,y_0)$? I think of Leibniz rule but its hypothesis include continuity in a neighborhood and not just at a point.
I appreciate any help.
analysis continuity partial-derivative
analysis continuity partial-derivative
asked Dec 2 at 4:08
Jiu
3409
asked Dec 2 at 4:08
Jiu
3409
asked Dec 2 at 4:08
Jiu
3409
asked Dec 2 at 4:08
Jiu
3409
asked Dec 2 at 4:08
Jiu
3409
3409
add a comment |
add a comment |
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