# Four Flavors of Continuity

Here's a chart to help keep track of some of the different "flavors" of continuity in real analysis.

# "Up to Isomorphism"?

Up to isomorphism” is a phrase that seems to get thrown around a lot without ever being explained. Simply put, we say two groups (or any other algebraic structures) are the same “up to isomorphism” if they’re isomorphic! In other words, they share the exact same structure and therefore they are essentially indistinguishable. Hence we consider them to be one and the same! But, you see, we mathematicians are very precise, and so we really don't like to use the word “same." Instead we prefer to say “same up to isomorphism.” Voila!

# Real Talk: Math is Hard, Not Impossible

The quote above comes from an excellent Slate article by Chase Felker on why students shouldn't be afraid of or intimidated by mathematics. I posted the quote on Instagram not too long ago, and since it addresses a topic that is near-and-dear to my own heart, I decided to include it on the blog as well. Felker prefaces the quote by saying, "Giving up on math means you don't believe that careful study can change the way you think."

# Transitive Group Actions: "Where There's a Will, There's a Way!"

In this post, we visually explore the definition of a transitive group action and see how it relates to the phrase, "Where there's a will, there's a way!"

# Dominated Convergence Theorem

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$, answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Dominated Convergence Theorem and see why "domination" is necessary.

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$ answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Monotone Convergence Theorem and solve a nasty-looking problem which, thanks to the theorem, is actually quite trivial!