Buckle up. Real analysis pulls no punches.

In my previous article, *Let’s Derive the Power Rule from Scratch!*, I ended the proof by citing the Moore-Osgood theorem to justify switching limits to prove the power rule for irrational powers. While I think it’s fair game to cite basic theorems, I’m a little unhappy with how I pulled out an advanced theorem from real analysis without explanation, as it goes against the “from scratch” spirit. In this article, we’re going to prove the Moore-Osgood theorem from scratch. In doing so, we’ll end up diving deep into the foundations of calculus and…

How can a sequence of continuous functions converge to a discontinuous function?

As with many fields, calculus started with a bunch of intuition and tricks. Over the next few centuries, mathematicians worked to justify their intuition and tricks by adding rigor to the field. In doing so, they found a lot of seemingly reasonable statements to be false, including

- switching the order of limits never changes the result,
- continuous functions must be differentiable except at a few points,
- all bounded functions are Riemann-integrable,
- and the limit of a sequence of continuous functions must be continuous.

Resolving these issues required refining…

You’ve seen limits in Calculus class and you know it has something to do with approaching, but how would you use it in a proof?

If you’re reading this article, you likely have a good, intuitive understanding of limits. The limit of *f(x)* as *x* approaches *a *is the value *f(x)* approaches when as *x* gets closer to *a*. In a more general sense, as the input approaches a value, the function approaches a limiting value.

While this intuition is nice and all, it won’t cut it in a proof. We need a precise definition of what it means to…

With nothing but a good model, a few definitions, and some math, we’ll derive a fundamental relationship of chemistry.

Anyone who has ever taken a chemistry class has seen the Ideal Gas Law:

We misunderstand entropy because we’re taught it’s a measure of disorder, but the real definition of entropy is far more useful and intuitive.

Look, I get it. You can’t throw a textbook at someone with no formal understanding of statistical mechanics when they ask a question. An answer that someone cannot understand is worth as much to them as no answer at all. Even taking this fact into account, **you shouldn’t give someone a wrong answer because you think they can’t understand the right answer**.

In this article, I’m going to explain entropy and how it has nothing to do…

How pop-science can lead to science denial.

In appealing to the general public, pop-science must simplify science. While science communicators must often leave out the math in their simplifications, they can go too far and leave out other important information. Worse, some of these simplifications change the facts, giving false information undue credibility. Pseudoscientists can then exploit this false information to then push their narrative. In this article, I want to talk about a case of simplification gone wrong — Creationism and Entropy.

While they may add their own spin on their entropy argument, the base of the argument is…

From nothing but the definitions of derivatives and limits, we’ll prove one of the first derivative rules you’ll learn in undergrad calculus.

Becoming a master at physics requires a lot of work, but learning the techniques and guidelines in this article is taking the first step.

People often talk about physics as if it were merely a set of facts and equations: ** F=ma**, energy can neither be created nor destroyed,

I am only going to talk about one commonly proposed solution to the excessive use of `if`

statements in the code of *Yandere Simulator* and nothing else about its development or any controversy surrounding it.

*Yandere Simulator *is filled to the brim with `if else`

chains (sections of code in which the computer goes down a list of conditions to check and executes the code for the first satisfied condition) and extremely nested `if`

statements (`if`

statements inside `if`

statements inside `if`

statements, etc.), both of which are bad practice.

*In this article, we’re going to explain how your computer uses memory and how you can work with your computer to write code that runs fast. To be clear, using memory properly will not make poorly written code easy to understand nor will it make a bad algorithm a good algorithm. Understanding how memory works, however, will help you find potential bottlenecks in your program and give you a way to solve them.*

I once had to clean up some poorly written code for a job I had. When I say poorly written, I don’t mean something minor like it…

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.