# Is the derivative of a discontinuity a delta function?

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In summary, the notes mention that in the case of V(x) containing delta functions, ψ'' also contains delta functions. This means that ψ' has finite discontinuities, and the derivative of a discontinuity can be thought of as a Dirac delta function times the step amount. However, this only applies to functions with simple step discontinuities and not other kinds of discontinuities. The concept can be made mathematically rigorous by defining "distributions".

#### Phys12

In these notes, https://ocw.mit.edu/courses/physics...-2016/lecture-notes/MIT8_04S16_LecNotes10.pdf, at the end of page 4, it is mentioned:

(3) V(x) contains delta functions. In this case ψ'' also contains delta functions: it is proportional to the product of a continuous ψ and a delta function in V. Thus ψ' has finite discontinuities.

If ψ' has finite discontinuities and ψ'' has delta functions, does that mean that the derivative of a discontinuity is a delta function?

Using informal mathematics, you can think of the "derivative" of a step function being the Dirac delta function times the step amount (including sign). A function with a simple step at a point ("simple" meaning that otherwise it would be differentiable) can be thought to have a derivative with a Dirac delta term at that point. There are other kinds of discontinuities that do not work that way.

This can be made mathematically rigorous by defining "distributions".

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Phys12

## 1. What is a delta function?

A delta function, also known as the Dirac delta function, is a mathematical concept used in calculus to represent a function that is zero everywhere except at a single point, where it is infinite. It is often used to model point sources in physics and engineering.

## 2. Is the derivative of a discontinuity always a delta function?

No, the derivative of a discontinuity is not always a delta function. It depends on the specific type of discontinuity and the function being differentiated. In some cases, the derivative may be a different type of singularity or may not exist at all.

## 3. Can a delta function be integrated?

Yes, a delta function can be integrated, but the result depends on the context in which it is being used. In some cases, the integral may be undefined or may require special techniques to evaluate.

## 4. Is a delta function a function in the traditional sense?

No, a delta function is not a function in the traditional sense, as it is not defined by a set of values at different points. Rather, it is defined by its properties and can be thought of as a distribution or generalized function.

## 5. What is the practical application of using a delta function in calculus?

The delta function is commonly used in physics, engineering, and other fields to model point sources or to simplify calculations involving discontinuous functions. It is also used in the theory of distributions, which has applications in differential equations and signal processing.