Let $C \subset \mathbb{R}^n$ be a compact, convex subset of $\mathbb{R}^n$. Let $f: C\to C$ be a continuous map that's not the identity map (see Qiaochu Yuan's comment below). If necessary here, assume differentiable, smooth, analytic... etc. We know by Brouwer's fixed point theorem that $f$ must have a fixed point.
My questions are:
(1) Are the fixed points of $f$ isolated even if $f$ is smooth, real analytic etc.? I understand that fixed points are isolated is equivalent to saying that the fixed points are finite.
(2) If not, is there a condition on $f$ that guarantees that the fixed points of $f$ are isolated?
EDIT: for question (1), we can use the examples of rotation around a fixed axis in a ball, as given by Qiaochu Yuan below; another example is:
$$f:B^2\to B^2:= f(x,y):=(x,y^2)$$
Note that here $x^2 + y^2 \le 1 \implies y^2 \le 1 \implies y^4 \le y^2 \implies x^2 + y^4 \le 1.$ Now here $f(x,y)=(x,y) \implies y^2=y \implies y= 0,1 \implies $ the fixed points of $f$ are the set of all $(x,0)$ and srt of all $(x,1), 0\le x \le 1.$ So the fixed point set is not isolated.