Reading: Read sections (30,) 31, 32
Exercises:
Let \(f(x) = x^2\) when \(x\) is rational and \(f(x) = 0\) otherwise.
Show that \(f\) is continuous at \(0\).
Show that \(f\) is discontinuous at all \(x \ne 0\).
Prove that \(f\) is differentiable at \(0\). (It is insufficient to simply claim \(f'(x) = 2x\))
Suppose \(f\) is differentiable at \(a\).
Prove that \(\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} = f'(a)\).
Prove that \(\lim_{h\to 0}\frac{f(a+h) - f(a-h)}{2h} = f'(a)\).
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 \(x = 0\).