I want to be able to find the discontinuities of a function and their respective "jumps".
My Idea would be to ask wolframalpha for the function of pointwise leftside limits, F-(x) = lim(y->x-) F(y), could someone please help me find the syntax necessary for doing so?
Given eg. the function F(x) = Piecewise[{{0, x < 0},{x, x >= 0 && x < 0.5}, {1, x >= 0.5}}], which has a discontinuity at x = 0.5 with a jump of 0.5.
How can I get Wolframalpha to give me the function F(x) - F-(x) = Piecewise[{{0, x < 0.5},{0.5, x = 0.5}, {0, x > 0.5}}]?
The actual problem I have:
I am given a distribution function F in stringified form and I need to find the atoms and their probabilities.
Now, there can be countably infinite many atoms, so enumerating those is not possible. My next idea was to return those atoms as a distribution function of those atoms, which is basically the "jumps" mentioned earler and where they occur