continuous function Archives - Infinity is Really Big

October 29, 2019October 29, 2019

Differentiable Functions and Local Linearity

Calculus 1, Lectures 12 through 15B

An infinitely oscillating continuous and differentiable function. — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

x=0 — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

In Steven Strogatz’s excellent book, Infinite Powers, there is a big emphasis on an idea he calls The Infinity Principle.

The concepts of a continuous function and of a differentiable function are … Read the rest

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.

… Read the rest