- #1
g1990
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Homework Statement
f(z) is a complex function (not necessarily analytic) on a domain D in C. The directional derivative is Dwf(z0)=lim(t->0) (f(z0+tw)-f(z0))/t, where w is a unit directional vector in C. There are three parts to the question:
a. Give an example of a function that is not differentiable at any point but that has directional derivatives is very direction w at every point z0. Very that your example satisfies the required properties
b. If f is differentiable at z0, show that there is a constant c such that Dwf(z0)=cw for every w
c.Assume that the real and imaginary parts of f has continuous partials. Show that if there exists a constant c such that Dwf(z0) exists and equals cw for every w, then f is differentiable at zo.
Homework Equations
Cauchy Riemann: du/dx=dv/dy and du/dy=-dv/dx
The Attempt at a Solution
for a, I know that my function can't satisfy the CR eqns, b/c it can't be differentiable, but I don't know how to plug a specific function into my directional derivative definition