Identity, Maths Club of IISER Kolkata

... or How I Learnt to Stop Worrying and Accept Mathematical Rigour. In Measure Theory, among other things, we try to generalize the concept of 'volume' (or 'area' or 'length' according as which one you prefer). In other words, like some sets come with an inherent 'measure' associated with them (for example, intervals in $\mathbb R$ comes with a length and "well behaved" shapes come with an area or volume depending on how many dimensions you count), we want to assign to any general subset a measure of its own. Since any general subset may not look as structured as the other sets that have their own volume, ideally we are free to randomly associate any number we want to to the set. But, as in all of mathematics, we want these numbers to satisfy certain properties. So, we will impose on these numbers a few properties that are very intuitive and geometric, but to our utter shock, are inconsistent. These properties are so much in tune with our ever...

A couple of days ago, Rohan Didmishe shared this problem with us: show that the function defined by \[ f\colon \mathbb{R} \to \mathbb{R}, \qquad f(x) = \begin{cases} x + x^2\sin(1 / x), &\text{ if }x \neq 0, \\ 0, &\text{ if } x = 0. \end{cases} \] is not monotonic (increasing or decreasing) in any interval $(-\delta, \delta)$ around zero. Graphing this function (say, using Desmos ) shows that it oscllates rapidly, curving up and down with increasing frequency the closer its gets to zero. This is due to the $x^2\sin(1 / x)$ term; the $x$ added in front 'tilts' the curve upwards. The first thing to look at is the derivative of $f$. Using $\lim_{x \to 0} x\sin(1 / x) = 0$ and the chain rule, we can compute \[ f'(x) = \begin{cases} 1 + 2x\sin(1 / x) - \cos(1 / x), &\text{ if }x \neq 0, \\ 1, &\text{ if } x = 0. \end{cases} \] Specifcally, $f'(0) = 1$ which seems to tell us that $f$ is increasing at $0$ ... or doe...