Real Analysis

What Is Real Analysis?

Whereas calculus focuses on continuous change, studied using derivatives and integrals, real analysis studies not just change, but all aspects of reals, real functions, and real sequences.

Example: Approximating Complicated Functions by Simpler Ones

Here's an example of something most people would consider part of real analysis, but not really a part of calculus. The Weierstrass approximation theorem says that given a continuous function $f(x)$ on a closed interval like $[0,1]$, you can find a polynomial $P(x)$ that approximates $f(x)$ as well as you like, in the sense that for any "precision challenge" $\epsilon>0$, you can choose the polynomial so that $|f(x) - P(x)| < \epsilon$ for all $x$ in the interval $[0,1]$. (So $P(x)$ "meets the $\epsilon$-precision challenge".)

The point is, even though $f(x)$ could be some really weird continuous function that you can't write down explicitly, if you're OK with an approximation (up to a small error of $\epsilon$), then you can pretend $f(x)$ is almost a polynomial. Since we understand polynomials much better than random functions, this is a big win!

Expii Choice Award in Literature: 101 Analysis Bedtime Stories

If you're curious to learn more, I highly recommend starting with Chapter 1, "epsilon-Red Riding Hood and the Big Bad Bolzano-Weierstrass Theorem" of the best intro to real analysis possible: 101 Illustrated Analysis Bedtime Stories.

Or, take a look at a more recent reboot: 101 Illustrated Real Analysis Bedtime Stories.