Recent content by CrumpledPaper

My bad. You're right =)

Good point, although in this case I don't think f(x)=1 is a valid counterexample because the norm is not one in \mathbb{C} for all points along the line (i.e. (1 + 100i)). A better counterexample would be a function defined by any single point on the circle.

Excellent. Thanks.

I don't seem to be getting anywhere still. Here's the complete problem: Let the function f: \mathbb{R} \rightarrow \mathbb{C} be defined by f(x) = \cos x + i \sin x for each real x. If a + ib is a complex number, then its absolute value is \left|a + ib\right| = (a^2 + b^2)^{\frac{1}{2}}; it...

What would constitute a rigorous proof for this problem?

The question examines f:\mathbb{R} \rightarrow \mathbb{C} tiny-tim, thanks for your help. I understand the "unit" portion of the proof. My difficulty is precisely in understanding how to prove that the function takes on all possible values. I thought a circle could be defined as x(t) = \cos t...

That's strange. What am I missing?

I stumbled across this question while self-learning topology, and I am not sure how to proceed. I need to prove that the image of f(x)=\cos x + i \sin x is the unit circle. The statement is obvious, which makes it all the harder for me to figure out what I actually need to do!