I know that $0^0$ is not generally defined to be any value since one of the reasons is that $z=x^y$ is discontinuous at $(x,y)=(0,0)$.
So I wanted to plot the graph of $z=x^y$ by myself using Matlab and I wrote the following Matlab code.
%% Create mesh grid
eps = 0.1; resolution = 100;
u = linspace(eps,2.0,resolution);
v = linspace(-2.0,2.0,resolution);
[x,y] = meshgrid(u,v);
%% Define z=x^y
z = x.^y;
mesh(x,y,z); xlabel('x'); ylabel('y'); zlabel('z');
So the result is my $z=x^y$ plot
But it doesn't make any sense since the shown graph does not reveal any discontinuity at $(x,y)=(0,0)$.
Of course I made a regularization because of singularity at the origin. But Wolfram alpha web service shows different result. Below is the result.
