Is the CDF of a Laplace distribution well-defined? So I'm asked to argue whether the CDF of a Laplace distribution is well-defined or not. Now I don't completely understand what well-defined actually means.
CDF given by:

Acording to wikipedia: "A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input"
So should I just input a bunch of of differently represented values. Like 0.5, 1/2 and pi for x and show the results? How would I show this function is well-defined or not?
 A: If well-defined can be interpreted as a true CDF, then you simply need to show that the function satisfies all the requirements of a CDF. The CDF as you've described is:
$$F(x)=\begin{cases}
\tfrac{1}{2}\text{exp}(\frac{x-\mu}{b})\,\quad\quad\,\,\,,\quad x<\mu\\
1-\tfrac{1}{2}\text{exp}(-\frac{x-\mu}{b})\,,\quad x\geq\mu\\
\end{cases}$$
Now, to be a CDF the function above has to be non-decreasing, càdlàg and also satisfy:
$$F(-\infty)=0\quad\text{and}\quad F(\infty)=1$$
For the function above, you should consider the two cases ($x<\mu$ and $x\geq \mu$) separately. To assess if they are non-decreasing, it suffices to show that the first derivative with respect to $x$ is non-negative:
$$\frac{d}{dx}F(x)\geq 0$$
To satisfy the càdlàg requirements, you can just show that the functions are continuous everywhere. You'll note that at $x=\mu$, the function is right-continuous. That just leaves that matter of finding the limits of the function, which is a simple enough task to prove.
For your interest, the form of the function is as follows (shown here with $\mu=b=1$ and $x\in[-10,10]$):

For your information, plotting the function is simple. Here is some MATLAB code to achieve this:
%Set values for constants:
mu=1;
b=1;

%Define two domains:
x1=-10:0.1:mu;  %x<mu
x2=mu:0.1:10;   %x>=mu

%Define the function for each x:
y1=0.5*exp((x1-mu)/b);
y2=1-0.5*exp(-(x2-mu)/b);

%Plot the function:
figure
set(gcf,'color','w')
plot(x1,y1)
hold on
plot(x2,y2)
hold on
plot(x2(1),y2(1),'r.','markersize',10)
xlabel('$x$','interpreter','latex');
ylabel('$F(x)$','interpreter','latex');

%Legend entries:
h=legend('$x<\mu$','$x\geq\mu$');
set(h,'Interpreter','latex')

A: The question here is not about whether this is a function or not (as in your Wikipedia citation), but rather about whether the function (which is unambiguously defined by the way) is actually a CDF (a non-decreasing, right-continous function, that has limit 0 at $-\infty$ and limit 1 at $+\infty$).
