Derivative of nonsmooth functions
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If a function is continuous but nonsmooth at point x = a, but has a finite derivative = 4 at x = a - $epsilon$ and a finite derivative = 7 at $x = a + epsilon$, both in the limit $epsilon to 0$.
Under what conditions (if any) is it appropriate to write that the derivative at x=a is $3 * delta(x - a)$?
Or do we say the derivative is infinity at $x = a$, or do we say the derivative is undefined at $x = a$?
derivatives
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
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add a comment |
$begingroup$
If a function is continuous but nonsmooth at point x = a, but has a finite derivative = 4 at x = a - $epsilon$ and a finite derivative = 7 at $x = a + epsilon$, both in the limit $epsilon to 0$.
Under what conditions (if any) is it appropriate to write that the derivative at x=a is $3 * delta(x - a)$?
Or do we say the derivative is infinity at $x = a$, or do we say the derivative is undefined at $x = a$?
derivatives
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
$endgroup$
1
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You can use the delta function to describe the derivative of a step function. Since f'(x) that has a step, it would be the second derivative that you might describe with a delta. The derivative itself is undefined at $a.$
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– Doug M
Dec 13 '18 at 17:37
add a comment |
$begingroup$
If a function is continuous but nonsmooth at point x = a, but has a finite derivative = 4 at x = a - $epsilon$ and a finite derivative = 7 at $x = a + epsilon$, both in the limit $epsilon to 0$.
Under what conditions (if any) is it appropriate to write that the derivative at x=a is $3 * delta(x - a)$?
Or do we say the derivative is infinity at $x = a$, or do we say the derivative is undefined at $x = a$?
derivatives
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
$endgroup$
If a function is continuous but nonsmooth at point x = a, but has a finite derivative = 4 at x = a - $epsilon$ and a finite derivative = 7 at $x = a + epsilon$, both in the limit $epsilon to 0$.
Under what conditions (if any) is it appropriate to write that the derivative at x=a is $3 * delta(x - a)$?
Or do we say the derivative is infinity at $x = a$, or do we say the derivative is undefined at $x = a$?
derivatives
derivatives
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
edited Dec 13 '18 at 16:44
Adeel Mahmood
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
asked Dec 13 '18 at 16:35
Adeel MahmoodAdeel Mahmood
113
113
1
$begingroup$
You can use the delta function to describe the derivative of a step function. Since f'(x) that has a step, it would be the second derivative that you might describe with a delta. The derivative itself is undefined at $a.$
$endgroup$
– Doug M
Dec 13 '18 at 17:37
add a comment |
1
$begingroup$
You can use the delta function to describe the derivative of a step function. Since f'(x) that has a step, it would be the second derivative that you might describe with a delta. The derivative itself is undefined at $a.$
$endgroup$
– Doug M
Dec 13 '18 at 17:37
1
1
$begingroup$
You can use the delta function to describe the derivative of a step function. Since f'(x) that has a step, it would be the second derivative that you might describe with a delta. The derivative itself is undefined at $a.$
$endgroup$
– Doug M
Dec 13 '18 at 17:37
$begingroup$
You can use the delta function to describe the derivative of a step function. Since f'(x) that has a step, it would be the second derivative that you might describe with a delta. The derivative itself is undefined at $a.$
$endgroup$
– Doug M
Dec 13 '18 at 17:37
add a comment |
2 Answers
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We can say (in the distributional sense) that the derivative of a "function" is $3delta(x-a)$ if there's a jump discontinuity of height $3$ at $x=a$.
In your case I would say it's more appropriate to simply say that the derivative is undefined there.
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
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With your assumptions you can say that the one-sided derivatives exist and are $= 4$ for the left side derivative and $=7$ for the right side derivative.
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
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2 Answers
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2 Answers
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$begingroup$
We can say (in the distributional sense) that the derivative of a "function" is $3delta(x-a)$ if there's a jump discontinuity of height $3$ at $x=a$.
In your case I would say it's more appropriate to simply say that the derivative is undefined there.
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
$endgroup$
add a comment |
$begingroup$
We can say (in the distributional sense) that the derivative of a "function" is $3delta(x-a)$ if there's a jump discontinuity of height $3$ at $x=a$.
In your case I would say it's more appropriate to simply say that the derivative is undefined there.
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
$endgroup$
add a comment |
$begingroup$
We can say (in the distributional sense) that the derivative of a "function" is $3delta(x-a)$ if there's a jump discontinuity of height $3$ at $x=a$.
In your case I would say it's more appropriate to simply say that the derivative is undefined there.
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
$endgroup$
We can say (in the distributional sense) that the derivative of a "function" is $3delta(x-a)$ if there's a jump discontinuity of height $3$ at $x=a$.
In your case I would say it's more appropriate to simply say that the derivative is undefined there.
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
answered Dec 13 '18 at 16:54
BigbearZzzBigbearZzz
8,69121652
8,69121652
add a comment |
add a comment |
$begingroup$
With your assumptions you can say that the one-sided derivatives exist and are $= 4$ for the left side derivative and $=7$ for the right side derivative.
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
$endgroup$
add a comment |
$begingroup$
With your assumptions you can say that the one-sided derivatives exist and are $= 4$ for the left side derivative and $=7$ for the right side derivative.
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
$endgroup$
add a comment |
$begingroup$
With your assumptions you can say that the one-sided derivatives exist and are $= 4$ for the left side derivative and $=7$ for the right side derivative.
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
$endgroup$
With your assumptions you can say that the one-sided derivatives exist and are $= 4$ for the left side derivative and $=7$ for the right side derivative.
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
answered Dec 13 '18 at 17:31
ThomasThomas
16.8k21631
16.8k21631
add a comment |
add a comment |
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You can use the delta function to describe the derivative of a step function. Since f'(x) that has a step, it would be the second derivative that you might describe with a delta. The derivative itself is undefined at $a.$
$endgroup$
– Doug M
Dec 13 '18 at 17:37