Reading: Read sections (30,) 31, 32
Exercises:
Let when is rational and otherwise.
Show that is continuous at .
Show that is discontinuous at all .
Prove that is differentiable at . (It is insufficient to simply claim )
Suppose is differentiable at .
Prove that .
Prove that .
Let \(f(x) = x \sin \frac 1x\) for \(x \ne 0\) and \(f(x) = 0\) for \(x = 0\). Note that \(f\) is continuous on \(\mathbb R\) (in fact, it is uniformly continuous on \([-R, R]\) for any \(R \ge 0\)).
Prove that \(f(x)\) is not differentiable at \(x = 0\).