Discontinuity Definition and 101 Threads

Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.
The oscillation of a function at a point quantifies these discontinuities as follows:

in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits of the two sides);
in an essential discontinuity, oscillation measures the failure of a limit to exist. The limit is constant.A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity).

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  1. J

    Roark's Equations for Discontinuity Stresses Syntax

    I am looking for clarification of some terms found in Roark's Formulas for Stress and Strain 9E. Table 13.4 pg 543 as well as preceding tables frequently reference R_A as the radius of common circumference. I take this to mean that this value could include any radius that both cylinders share...
  2. M

    Infinite discontinuity question

    For 6(b), The solution is, However, for ##a = 1## they could have also said that f is not continuous since f(1) is not defined (vertical asymptote) correct? Many thanks!
  3. V

    Discontinuity in an Electric line of force

    This is a tricky and difficult question for me. I know from reading various textbooks that electric lines of force are always continuous without breaks, but cannot pinpoint a reason for this. The only reason I can come up is that an electric line of force must always begin and end on charges...
  4. chwala

    I Understanding Removable discontinuity of the given function: ##e^x##

    I would like to understand the highlighted part. In my understanding, this function does not seem to have a hole! Having said this, i can state that ##x_0=1## and we have our defined ##f(x_0)=2##. It follows that, ##f(1^{+}) = e## ##f(1^{-}) = e## thus ##f(x_0^{+})=f(x_0^{-})≠f(x_0)## thus the...
  5. A

    The value of a Fourier series at a jump point (discontinuity)

    Greetings according to the function we can see that there is a jump at x=e and I know that the value of the function at x=e should be the average between the value of f(x) at this points my problem is the following the limit of f(x) at x=e is -infinity and f(e)=1 how can we deal with such...
  6. hilbert2

    A Softened potential well / potential step

    Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length ##2L## ##V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.## and a differentiable one ##\displaystyle V(x) = V_0...
  7. Leo Liu

    Discontinuity of velocity in a problem on mass flow and momentum

    In this question, the particles are constantly transmitting their momentum to the rocket. The force required to keep the rocket stable can be express as ##\vec F=(\vec v-\vec u)\dot m##. However, when I tried to solve this question using the Newton's 2nd law, I found that the infinitesimal...
  8. karush

    MHB S8.02.4.58 type of discontinuity tikx with 1/sqrt(1-cos(2x))

    $\tiny{s8.2.4 58}$ desmos Let $f(x)=\dfrac{1}{\sqrt{1-\cos{2x}}}$ (a) Graph $f$ What type of discontinuity does it appear to have at 0?\\ (b) Calculate the left and right limits of $f$ at 0. \\ Do these valuesn' confirm your answer to part (a)? doesn't the limit going to 0 infinity both...
  9. K

    I Riemann integrability with a discontinuity

    So, I know that a function is integrable on an interval [a,b] if ##U(f,P_n)-L(f,P_n)<\epsilon ## So I find ##U(f,P_n## and ##L(f,P_n## ##L(f,P_n)=5(3-\frac{1}{n}-0)+5(3+\frac{1}{n}-(3-\frac{1}{n}))+7(4-(3+\frac{1}{n}))=22-\frac{2}{n} ##...
  10. G

    Discontinuity in the value of arctan(y/x)

    after two simple line integrals we find that $$ f(x,y) = \arctan{p/q} + \arctan{y/x} + \pi/2$$ if ##x > 0## and $$f(x,y) = \arctan{p/q} + \arctan{y/x} - \pi/2$$ if ## x < 0 ##. And then we can just take the limits to find the value of the discontinuity (as given above). But what is the...
  11. U

    I Boundary conditions and discontinuity of EM fields

    Premise: everything that follows is done in the frequency domain. Boundary conditions If there are superficial currents (electric and magnetic) impressed on the boundary between two media, we have these discontinuities for the tangential components of the fields...
  12. R

    MHB Finding Discontinuities: Multiply by Conjugate?

    I recently had to find what f(7) equals if f(x) = \frac{x^2-11x+28}{x-7}. I first tried \frac{x^2-11x+28}{x-7} \cdot \frac{x+7}{x+7}, and it seemed like a perfect fit since I eventually got to \frac{x^2(x-4)-49(x+4)}{x^2-49}=(x-4)(x+4), but that gave me f(7)=33, instead of the right answer...
  13. M

    What will the electric field be at the surface?

    The electric field due to a dipole distribution in volume ##V'## can be viewed as electric field due to a volume charge distribution in ##V'## plus electric field due to a surface charge distribution in boundary of ##V'##. ##\displaystyle\mathbf{E}=\int_{V'} \dfrac{\rho...
  14. P

    I Is the derivative of a discontinuity a delta function?

    In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-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...
  15. G

    Calculus-I: discontinuity of piecewise function HW problem

    Screenshot of my homework problem along with my solution so far. I'm not sure if I'm doing this correctly and if I am... if I'm answering correctly. Thank you. (EDIT: I made 1 small error with the piecewise definition. Ignore the f(x) before g(x).)
  16. D

    Is the Parenago Discontinuity a real thing?

    Recently I came across a discussion on a Pantheist forum concerning the movement of cooler Stars & the Parenago Discontinuity. The proposal was that this disconinuity could be used as a way of testing whether cooler Star have self determination... yeah yeah I know, I'm not asking here for a...
  17. amjad-sh

    Discontinuity in electric field

    I was recently reading jackson's book of electrodynamics regarding surface charge distributions and discontinuity in electric field. I reached that part and almost didn't understand anything. I will write the whole text: One of the common problems in electrostatics is the determination of...
  18. Schaus

    Finding discontinuities in functions

    Homework Statement Where are the following functions discontinuous? f(x) = (x+2)/√((x+2)x) Homework EquationsThe Attempt at a Solution f(x) = (x+2)/√((x+2)x) = (x+2)/x√(2x) multiply both denominator and numerator by √(2x) = (x√2+2√x)/(x(2x)) Can I leave it like this and state that x ≠ 0, or...
  19. Jehannum

    I What Defines a Piecewise Smooth Function?

    For example, if we define f(x) as "the greater of x and x2" it will give a straight line graph between (0,0) and (1,1) then turn into a curve. This function is continuous but not 'smooth'. Is there any special name for this kind of function? Are there any interesting considerations about such...
  20. C

    Proving discontinuity for rational numbers (reduced form)

    Hello! This is my first post on these forums. So I was stuck with this question in my Mathematical Analysis exam, and it is as follows: ƒ(x) = 0 if x ∉ ℚ and (p + π) / (q + π) - (p / q) if x = (p / q) ∈ ℚ (reduced form). 1- Prove ƒ is discontinuous at all rational numbers except 1: This is...
  21. A

    B How can we measure Fragmentation/ discontinuity ?

    How can we measure Fragmentation/ discontinuity in a system?
  22. twist.1995

    Electric field discontinuity in a spherical shell?

    Homework Statement Lets say, there is a non-uniform charge distribution, given as in a spherical shell that has a cavity with radius a and the radius b to the outer surface. I am wondering if the field is discontinuous just on the surface of this sphere. Homework Equations...
  23. S

    MHB Discontinuity: Asymptotic discontinuity

    Hi again! I thought I would finish off my previous posts on discontinuity by discussing asymptotic discontinuity. So let's focus on these alone in this thread. I'm not familiar with the origin of the term "asymptote", but from what I can tell, it has asymptotic discontinuity to thank for for...
  24. S

    MHB Discontinuity: Jump discontinuity

    I earlier posted about point discontinuity. It became overwhelming pretty quickly. Now I would like to focus this thread at jump discontinuity specifically, if you don't mind me posting multiple threads about discontinuity. From what I understand, "jump discontinuity" occurs where the left-hand...
  25. S

    MHB Discontinuity: Point discontinuity

    At the very basic level, I understand this notion of discontinuity. But I am looking to understand this better, at a deeper level. Because I know I will need to know this well as I start to explore piece-wise functions. Why do we bother drawing graphs? I want to be able to tell if a function...
  26. S

    What qualifies as an infinite discontinuity?

    Hello everyone. I am currently having trouble actually defining what qualifies as an infinite discontinuity. I have read several sources that state that both of the one sided limits must approach infinity (positive, negative or both). My problem is what happens when only one of the one sided...
  27. S

    Need Help With Discontinuity Proof

    Let D⊆ℝ be an interval of nonzero length from which at most finitely man points x1,...,xn have been removed and let f: D→ℝ be a function. Then every discontinuity x∈D∪{x1,...,xn} of f is one of four types (removable, jump, infinite or discontinuity by oscillation). Proof: Let x∈D or let x be...
  28. S

    Vector Potential A: Discontinuity at the surface current

    Homework Statement Prove Eqn. 1 (below) using Eqns. 2-4. [Suggestion: I'd set up Cartesian coordinates at the surface, with z perpendicular to the surface and x parallel to the current.] Homework Equations I used ϑ for partial derivatives. Eqn. 1: ϑAabove/ϑn - ϑAbelow/ϑn = -μ0K Eqn. 2: ∇ ⋅ A...
  29. kostoglotov

    Discontinuity of a constraint in Lagrange Method

    Homework Statement My question is quite specific, but I will include the entire question as laid out in the text Consider the problem of minimizing the function f(x,y) = x on the curve y^2 + x^4 -x^3 = 0 (a piriform). (a) Try using Lagrange Multipliers to solve the problem (b) Show that the...
  30. D

    Calculus Question on Removable Discontinuity

    I don't entirely understand the question which is why I am posting it here. Anyways, from what the question is asking;we are trying to find the removable discontinuity. This would be plugging in x=0 into both equation and combining them. When this is applied to the first equation, the answer is...
  31. P

    Normal derivative of vector potential discontinuity

    Homework Statement In Griffiths, the following boundary condition is given without proof: ∂Aabove/∂n-∂Abelow/∂n=-μ0K for the change in the magnetic vector potential A across a surface with surface current density K, where n is the normal direction to the surface. A later problem asks for a...
  32. L

    MHB How do I determine the type of discontinuity for a discontinuous function?

    I need help with discontinuous functions. More specifically, how to determine what type of discontinuity they are, algebraically. Example: Determine whether each function is continuous at the given x values. Justify using the continuity test. If discontinuous, identify the type of discontinuity...
  33. A

    MHB Graphing Functions With Asymptotes and Discontinuity

    Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: Sorry guys, I can't upload my sketched graph. But I can describe it: Given the four conditions, I got a dicontinuous graph that...
  34. O

    Function with a removable discontinuity

    Is a function with a removable discontinuity considered continuous? I've looked through about 6 calculus texts and none of them really go into any detail.
  35. G

    Does a limit exist at a point of discontinuity?

    Just wanted to clear up something : If there is a removable discontinuity on a graph(1,2) with both a right and left hand portion and let's say the question was : whether lim at x-> 1f(x) exist Would it be yes it does exist just that it is undefined or?
  36. G

    Removable discontinuity solution

    Problem statement Revelant equations None Attempt at a solution I know it is discontinuous if the right hand limit doesn't equals the left hand limit? Is that correct? The other criteria are If f(c) exists, lim f(x) x--> c exists and lim f(x)=f(c) I don't really understand what the other...
  37. A

    Real Analysis: Continuity and discontinuity

    1) For the following choice of A, construct a function f: R → R that has discontinuities at every point x in A and is continuous on the complement of A. A = { x : 0 < x < 1} My function is f(x) = 10 if x in (0,1) and Q and f(x) = 20, if x in (0,1) and irrational number, f(x) = 30, elsewhere...
  38. N

    What is the Point Discontinuity Problem in Rational Functions?

    Homework Statement Find the value of p at which the discontinuity would occur. Homework Equations f(x) = x^2 - 6x + 9 / x - p The Attempt at a Solution Able to solve if p has an assigned numerical value, but help is needed for determining the value of p at which the...
  39. T

    Discontinuity of electric field at surface of a conductor

    Hi all, So I've been reviewing for the PGRE at the end of next month (wish me luck, I'm going to need it) and I stumbled across something that confused me in my old textbook. I was reading about the discontinuity of the electric field at the surface of a conductor, and also about the...
  40. P

    Discontinuity in electrical field.

    Hi, Homework Statement I'd like to find the electrical field at the center of a hole made in a hollow sphere of radius R0 with uniform charge density σ. The radius of the disc-shaped hole is a << R0. Homework Equations I know that the electrical field of the sphere is Q/r2 for r>R and 0 for...
  41. N

    Integration over a discontinuity

    Homework Statement Evaluate ∫(1/x)sin^2(x)dx from -a to a The Attempt at a Solution Mathematica doesn't want to evaluate this because of the lack of convergence. I think it is zero. When we consider non-zero values of x in the associated riemann sum, the integrand is odd and so...
  42. S

    Discontinuity Waves: Deriving the expressions

    Homework Statement A long string of linear mass density μ_1 = 1.0 gr/cm is joined to a long string of linear mass density μ_2 = 4.0 gr/cm and the combination is held under constant tension. A transverse sinusoidal wave of amplitude A _i = 3.0 cm and wavelength λ =25 cm is launched...
  43. J

    Fourier series of functions with points of discontinuity

    If you have a function with countable discontinuities on an interval, I know that the Fourier series will converge to that function without those discontinuities. But how could you explain that formally? If the basis of the Fourier series span the space L^2[a,b], that would include functions...
  44. D

    MHB What are the points of discontinuity for the function $f(x,y)$?

    Points of discontinuity $f(x,y) = \begin{cases}\frac{\sin x - \sin y}{\tan x - \tan y}, & \text{if } \tan x\neq\tan y\\ \cos^3 x, & \text{if } \tan x = \tan y\end{cases}$ Not sure what to do with this one.
  45. J

    Find the points of discontinuity: f(x) = x + 1 , for x < 1 and 1/x for x ≥ 1?

    Find the points of discontinuity: f(x) = x + 1 , for x < 1 and 1/x for x ≥ 1? ^ supposed to be a piece-wise function. State whether f is left- or right-continuous at each point of discontinuity. I'm having difficulty figuring this out... please help?
  46. I

    Finding removable discontinuity

    Homework Statement Let f(x)= {x^2-7x+10, for x^2 ≠ 25 { A, for x = 5 { B, for x = -5 Is there a value of A that makes f continuous at x= 5? Is there a value of B that makes f continuous at x= -5?
  47. mesa

    Need a function with a jump discontinuity

    What would be a good example of a function that would give a jump discontinuity without using piecewise or absolute values?
  48. A

    How to Implement FEM with a Discontinuity in the Exact Solution?

    Homework Statement Use FEM to solve this problem. The difficulty lies in the fact that the exact solution has a discontinuity in it. From x=[0,0.6) the exact solution u is x5/20 - x/20 and from x =(0.6,1] u is sin(x). The problem I'm having is I'm not sure what to do at the jump in my code...
  49. E

    Parallel plate waveguide, step discontinuity

    Hello! You can find in the picture attached a parallel plate waveguide which has an a1 height before the step and an a2 height after the step. The plates are perfect conductors and the step is ideal. I can't determine which physical quantities are continuous across this discontinuity. Suppose...
  50. C

    Energy conservation in electric field scattering from a discontinuity

    Hi, I'm a EE PhD student working a little bit out of my area, and have just gotten stumped trying to figure out the transient dymanics of a relativistic electron moving past a discontinuity. My little thought problem came up from wakefield interactions of a relativistic electron in a waveguide...
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