Continuity and Differentiability of Step function?
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All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How is this possible??
derivatives continuity dirac-delta step-function
asked Nov 14 at 17:24
robin
183
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up vote
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All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How is this possible??
derivatives continuity dirac-delta step-function
asked Nov 14 at 17:24
robin
183
The Dirac Delta is not a function. It is a distribution (in the sense of the Theory of Distributions). What would be the value of $delta(0)$ ??
– Yves Daoust
Nov 14 at 17:38
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How is this possible??
derivatives continuity dirac-delta step-function
asked Nov 14 at 17:24
robin
183
All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How is this possible??
derivatives continuity dirac-delta step-function
derivatives continuity dirac-delta step-function
asked Nov 14 at 17:24
robin
183
asked Nov 14 at 17:24
robin
183
asked Nov 14 at 17:24
robin
183
asked Nov 14 at 17:24
robin
183
asked Nov 14 at 17:24
robin
183
183
The Dirac Delta is not a function. It is a distribution (in the sense of the Theory of Distributions). What would be the value of $delta(0)$ ??
– Yves Daoust
Nov 14 at 17:38
add a comment |
The Dirac Delta is not a function. It is a distribution (in the sense of the Theory of Distributions). What would be the value of $delta(0)$ ??
– Yves Daoust
Nov 14 at 17:38
The Dirac Delta is not a function. It is a distribution (in the sense of the Theory of Distributions). What would be the value of $delta(0)$ ??
– Yves Daoust
Nov 14 at 17:38
The Dirac Delta is not a function. It is a distribution (in the sense of the Theory of Distributions). What would be the value of $delta(0)$ ??
– Yves Daoust
Nov 14 at 17:38
add a comment |
2 Answers
2
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My guess is that you are calling “step function” to a function like$$begin{array}{rccc}scolon&mathbb R&longrightarrow&mathbb R\&x&mapsto&begin{cases}0&text{ if }x<0\1&text{ otherwise.}end{cases}end{array}$$Yes, it is a function and, yes, it is not continuous (at $0$). But it is not differentiable (again, at $0$). It is oftain said that $s'$ is the Dirac function, but that is only an idea about what the Dirac function is. Actually, the limit $displaystylelim_{xto0}frac{s(x)-s(0)}x$ doesn't exist (in $mathbb R$) and therefore $s$ is not differentiable at $0$.
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
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up vote
2
down vote
Your initial thesis is incorrect: step functions are not differentiable at the points of discontinuity, and the Dirac Delta is not a function.
answered Nov 14 at 17:30
user3482749
1,088411
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
My guess is that you are calling “step function” to a function like$$begin{array}{rccc}scolon&mathbb R&longrightarrow&mathbb R\&x&mapsto&begin{cases}0&text{ if }x<0\1&text{ otherwise.}end{cases}end{array}$$Yes, it is a function and, yes, it is not continuous (at $0$). But it is not differentiable (again, at $0$). It is oftain said that $s'$ is the Dirac function, but that is only an idea about what the Dirac function is. Actually, the limit $displaystylelim_{xto0}frac{s(x)-s(0)}x$ doesn't exist (in $mathbb R$) and therefore $s$ is not differentiable at $0$.
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
add a comment |
up vote
0
down vote
accepted
My guess is that you are calling “step function” to a function like$$begin{array}{rccc}scolon&mathbb R&longrightarrow&mathbb R\&x&mapsto&begin{cases}0&text{ if }x<0\1&text{ otherwise.}end{cases}end{array}$$Yes, it is a function and, yes, it is not continuous (at $0$). But it is not differentiable (again, at $0$). It is oftain said that $s'$ is the Dirac function, but that is only an idea about what the Dirac function is. Actually, the limit $displaystylelim_{xto0}frac{s(x)-s(0)}x$ doesn't exist (in $mathbb R$) and therefore $s$ is not differentiable at $0$.
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
My guess is that you are calling “step function” to a function like$$begin{array}{rccc}scolon&mathbb R&longrightarrow&mathbb R\&x&mapsto&begin{cases}0&text{ if }x<0\1&text{ otherwise.}end{cases}end{array}$$Yes, it is a function and, yes, it is not continuous (at $0$). But it is not differentiable (again, at $0$). It is oftain said that $s'$ is the Dirac function, but that is only an idea about what the Dirac function is. Actually, the limit $displaystylelim_{xto0}frac{s(x)-s(0)}x$ doesn't exist (in $mathbb R$) and therefore $s$ is not differentiable at $0$.
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
My guess is that you are calling “step function” to a function like$$begin{array}{rccc}scolon&mathbb R&longrightarrow&mathbb R\&x&mapsto&begin{cases}0&text{ if }x<0\1&text{ otherwise.}end{cases}end{array}$$Yes, it is a function and, yes, it is not continuous (at $0$). But it is not differentiable (again, at $0$). It is oftain said that $s'$ is the Dirac function, but that is only an idea about what the Dirac function is. Actually, the limit $displaystylelim_{xto0}frac{s(x)-s(0)}x$ doesn't exist (in $mathbb R$) and therefore $s$ is not differentiable at $0$.
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
answered Nov 14 at 17:33
José Carlos Santos
140k19111204
140k19111204
add a comment |
add a comment |
up vote
2
down vote
Your initial thesis is incorrect: step functions are not differentiable at the points of discontinuity, and the Dirac Delta is not a function.
answered Nov 14 at 17:30
user3482749
1,088411
add a comment |
up vote
2
down vote
Your initial thesis is incorrect: step functions are not differentiable at the points of discontinuity, and the Dirac Delta is not a function.
answered Nov 14 at 17:30
user3482749
1,088411
add a comment |
up vote
2
down vote
up vote
2
down vote
Your initial thesis is incorrect: step functions are not differentiable at the points of discontinuity, and the Dirac Delta is not a function.
answered Nov 14 at 17:30
user3482749
1,088411
Your initial thesis is incorrect: step functions are not differentiable at the points of discontinuity, and the Dirac Delta is not a function.
answered Nov 14 at 17:30
user3482749
1,088411
answered Nov 14 at 17:30
user3482749
1,088411
answered Nov 14 at 17:30
user3482749
1,088411
answered Nov 14 at 17:30
user3482749
1,088411
1,088411
add a comment |
add a comment |
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The Dirac Delta is not a function. It is a distribution (in the sense of the Theory of Distributions). What would be the value of $delta(0)$ ??
– Yves Daoust
Nov 14 at 17:38