Consider the real-valued function f defined on the real numbers as
f(x)={e−1/x2,0,when x=0;when x=0.
((a)) Show that, for all n>0, f(n)(0)=0, where f(n) is the nth derivative of f. Hint: Use the limit definition for the derivative to establish the case for n=1 and then use mathematical induction to complete your argument.
((b)) Explain why the function f is an example of a function that, although differentiable everywhere on the real line, is not expressible as a Taylor series about 0 that is valid for any interval (−ε,ε), no matter how small ε is. Hint: Evaluate the Taylor series representation for f(x) when x=0, and show that the series does not equal f(x).
((c)) Explain why a similar argument could not be made for the complex-valued function g defined on the complex numbers as
g(z)={e−1/z2,0,when z=0;when z=0.
Hint: Show that g(z) is not even continuous at z=0 by taking limits along the real and imaginary axes.