Is exp(-bx) piecewise continuous on every bounded interval?

[jaejoon89]

Homework Statement

Verify that exp(-bx) where b is a positive constant satisfied the conditions of the Fourier integral theorem given in our book (see below).

Homework Equations

N/A

The Attempt at a Solution

The theorem says under what conditions the Fourier series applies. The conditions are that f must be absolutely integerable for x > 0 and piece wise continuous on every bounded interval on it. Also, f(x) at each point of discontinuity of f must be the mean value of the one-sided limits f(x+) and f(x-). f(x) represents the Fourier integral, and f represents the original function. The theorem says that if these conditions hold, you can write f as the Fourier series f(x).

I am supposed to verify the conditions mentioned above for e^(-bx) where b is a positive constant.

My question is how is exp(-bx) piecewise smooth on every bounded interval of it? It is not a piece wise function.

 

[Russell Berty]

It is "piecewise", it only has one piece, and that piece is continuous on all of R.