Articles of piecewise continuity

Matlab's “buggy” symbolic integration function seems to be equivalent to the Implied Integral. Coincedence or intentional?

In matlab there is a function called “int(f,x)” which takes in a symbolic function and a variable and integrates it. I decided to try using it today to play around and do a few integrals and I got a pretty interesting surprise. I integrated the following and got the following results from matlab: $$\int floor(x) […]

Turning a Piecewise Function into a Single Continuous Expression

I’m trying to construct a function that contains two parts: one, $g(a,b)$, coming into effect when $a\lt b$, and the other, $h(a,b)$, when $b \lt a$. The problem is that $a=b$ is a valid possibility which I do not know how to handle, even though if $a=b$, then $g(a,b)=h(a,b)$ and $g'(a,b)=h'(a,b)$. So there is a […]

Conjectures Involving floor(x)/piecewise continuity with regards to Integration and Differential Equations

[This answer has been heavily edited in response to a long chain of comments on Eric Stucky’s answer.] I came up with these few theorems and I am curious whether or not my hypothesis are true. I don’t really have a method for proving these, mind you. I’m more considering these as things I’ve noticed […]

Iterating through the integerial points of $f(x)$

The floor function $g(x)=\left\lfloor f(x) \right\rfloor$ jumps at points where $f(x)$ is an integer. What I want is a function that gives the $n$-th jump point from $g(0)$. So, for let’s say $g(x)=\left\lfloor x \right\rfloor$, those points would be $1,2,3,4…$ in that order. For negative values that would give jump points in the left half […]

Correlation between the weak solutions of a differential equation and implied differential equations

Yes,this is very similar to a previous question I asked. That was about normal solutions and not weak solutions. We define the operator known as the implied derivative denoted as $I(f)(x)(g)$ to be: $$I(f)(x)(g) := g(x) \left(\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} \right) + (1-g(x)) \left(\lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h} \right)$$ Where $g(x)$ is an arbitrary characteristic/indicator function. I wish […]