Exercise 1, § 1

Let $f$ be a vector function of a real variable, defined on an interval $I \subset \mathbf{R}$ and differentiable at a point $x_0$ interior to $I$. Show that the quotient

tends to $f'(x_0)$ as $h$ and $k$ tend to 0 through values $> 0$. Converse.

Show that the function $f$ equal to $x^2 \sin 1/x$ for $x \neq 0$, and to 0 for $x = 0$, is everywhere differentiable, but that $(f(y) - f(z))/(y - z)$ does not approach $f'(0)$ as $y$ and $z$ tend to 0, while remaining distinct and $> 0$.