I found that the statement about the cross entropy not true. When y==a the function is non-monotonic with 0 at the extremes but not at the middle. So the "proof" shown is confusing to me.
The statement "if the neuron's actual output is close to the desired output, i.e., y=y(x) for all training inputs x, then the cross-entropy will be close to zero"
is not true. The function peaks in the middle (~ 0.7)
Thanks! -Kaushik
The essential point is that we're considering classification problems, for which the output is intended to be 0 or 1. I address the more general case of regression problems (where y may take any value) in a later exercise.
Hope that helps!