Questions about Lipschitz function

Questions about Lipschitz function

Let $f : X \to Y$ be a function. We say f is Lipschitz if there exists a
constant C such that d(f(x), f(y)) $\leq$ Cd(x, y).
a) Prove $f(x) = x^{2}$ is not Lipschitz on R.
b) Prove that the function $f : R^{2} \to R$, where f(x) is the distance
from x to the unit circle, is Lipschitz.
c) If $f : [1, b] \to R$ is a differentiable $f$ whose derivative is
continuous. Prove that $f$ is Lipschitz.
Could you tell me if my solutions for a), b) are correct and help me solve
c)?
For a), I thought For $x_1, x_2$ $\in$ $X$, $|x_2^{2} - x_1^2| \leq
C|x_2-x_1|$ which means $|x_2+x_1| \leq C$. Since this should be true for
x1,x2 its impossible since C is a fixed value.
For b), I thought we can draw a triangle by linking a point x, a point y
and a center of unit circle. Then each line's length will be |f(x)|,
|f(y)| and d(x,y).
Hence, $d(x,y) > |f(x)| - |f(y)| = d(f(x), f(y))$ So there exists C(=1)
such that d(f(x), f(y)) $\leq$ Cd(x, y)
I have a no idea about c).