continuity Archives - Infinity is Really Big

June 14, 2021July 1, 2021

Existence of nth Roots of Positive Reals

The Completeness Axiom helps us prove the existence of nth roots of positive real numbers, but the proof is quite challenging.

Real Analysis Study Help for Baby Rudin, Part 1.6

The existence of nth roots, including the square root of two, is ultimately based on the continuity of power functions with positive integer exponents. — Ultimately, the continuity of $f(x)=x^{2}$ at $x=\sqrt{2}$ allows us to prove the existence of $\sqrt{2}$ . Later, the argument is based on the Intermediate Value Theorem. In this article, the argument is

f(x)=x^{2} — Ultimately, the continuity of $f(x)=x^{2}$ at $x=\sqrt{2}$ allows us to prove the existence of $\sqrt{2}$ . Later, the argument is based on the Intermediate Value Theorem. In this article, the argument is

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October 15, 2019October 15, 2019

Limit Definition, Continuity, and Derivatives

Calculus 1, Lectures 8A through 11

The limit definition of the derivative leads to the graph of a function with a hole (a removable discontinuity). The limit still exists, based on the precise epsilon delta definition of a limit. — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

DQ(h)=\frac{(2+h)^{2}-8}{h} — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

Graphs with holes (missing points) might seem like anomalies — like they are mere curiosities hardly worth studying. However, they arise at the heart of differential calculus.

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