Buckle up. Real analysis pulls no punches.

limit as x approaches a limit as y approaches b of f of x and y equals with a question mark limit as y approaches b limit as x approaches a of f of x and y equals with a question mark limit as (x, y) approaches (a, b) of f of x and y. Basically, does switching the order of limits or taking both at the same time affect the output?
limit as x approaches a limit as y approaches b of f of x and y equals with a question mark limit as y approaches b limit as x approaches a of f of x and y equals with a question mark limit as (x, y) approaches (a, b) of f of x and y. Basically, does switching the order of limits or taking both at the same time affect the output?

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?

A bridge with the cables converging in the center of the image.
A bridge with the cables converging in the center of the image.
Photo by Max Sandelin on Unsplash

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

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?

Photo by Tom Podmore on Unsplash

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.

A set of balloons.
A set of balloons.
Photo by Blessing Ri on Unsplash

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.

Photo by Luke Porter on Unsplash

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.

Photo by Johannes Plenio on Unsplash

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.

The Creationist Entropy Argument

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


Back to School

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

Photo Credit: Shubham Sharan on Unsplash

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

The author made all images in this article using LaTeX, tikz, numpy, pyplot, and GIMP.

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, E = mc², stars are giant balls of hydrogen fusing into helium and heavier elements, etc. These facts and equations can either form the base of our understanding or result from our understanding, but to do physics is to connect the two. …


Many people have rightly criticized Yandere Simulator’s overuse of if statements, but the issue is much more than switch statements can fix.

A character model from Yandere Simulator dressed as a graduate standing in front of an if else chain from the game’s code.
A character model from Yandere Simulator dressed as a graduate standing in front of an if else chain from the game’s code.
From u/Apterygiformes through r/ProgrammerHumor

Disclaimer

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.

The Context

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.

With the exception of one or two people I’ve seen…


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…

Joseph Mellor

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

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