write as a Fourier series:
b is a positive odd integer and
This is a jagged function that appears in models of Brownian motion. Each partial sum is continuous, and hence by the uniform limit theorem (which states that the uniform limit of any sequence of continuous functions is continuous), the series (1.6) is continuous.
1.2.4 Classes of Discontinuous Functions
A function that is not continuous at some point is said to be discontinuous at that point. For example, the Heaviside function (1.2) is not continuous at
There are two (simple discontinuity) main categories of discontinuous functions:
First kind: and exists. Then either we have or .
Second kind: a discontinuity that is not of the first kind.
Examples are:
You can check that this latter function has a discontinuity of the first kind at
1.3 DIFFERENTIAL CALCULUS
The derivative of a function is one of its fundamental properties. It represents the rate of change of the slope of the function: in other words, how fast the function changes with respect to changes in the independent variable. We focus on real-valued functions of a real variable.
Let
This limit may not exist at certain points, and it is possible to define right-hand and left-hand limits, that is, one-sided derivatives.
Some results that we learn in high school are:
(1.8)
A composite function is a function that we can differentiate using the chain rule that we state as follows:
(1.9)
