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Suppose the following functions:

Case 1:

$$y=f(x), x\in\mathbb{R}.$$

It can be evaluated anywhere on a specified interval $[a,b]$ and has no noise. How to detect the following discontinuities $c$ at a point $x=c$ (or at its boundary $x\pm\varepsilon$), where $\varepsilon\rightarrow0$:

  • removable discontinuities,
  • jump discontinuities,
  • infinite discontinuities.

In my opinion, last two discontinuities appear a straight line parallel to the $y-$ axis. I tried also to estimate the empirical error of Simpson's rule $$ E(f,c,c+\varepsilon)=\frac{1}{15}(S(f,c,c+\varepsilon/2)+S(f,c+\varepsilon/2,c+\varepsilon)-S(f,c,c+\varepsilon)), $$ where $$ S(f,a,b)=\frac{b-a}{6}(f(a)+4f(c)+f(b)),\quad c=0.5(a+b), $$ indicating the rapid increment/decrement of the area.

Case 2: System of parametric functions \begin{align*} x & =f(u,v),\\ y & =g(u,v), \end{align*} where $u\in[-\pi/2,\pi/2]\wedge v=v_{0}\vee u=u_{0}\wedge v\in[-\pi,\pi]$. Here, a parametric function may pass along the singularity $c$ if: \begin{align*} u & \in[-\pi/2,\pi/2]\wedge v_{0}=c\vee u_{0}=c\wedge v\in[-\pi,\pi] \end{align*} or intersect the singularity $c$ \begin{align*} u & =c\wedge v=v_{0}\vee u=u_{0}\wedge v=c. \end{align*}

How to detect all three discontinuities at a point $c$ (or at its boundary)?

Is there any computationally inefficient method working without the complex arithmetic?

Thanks for your help.

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