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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5
votes
Does there exist a continuous field of directions in $\mathbb R^3$ tangent to every sphere?
About the bonus question:
Just take your favorite function $f:\mathbb{R}^2\to \mathbb{R}$ without critical points, say for example $f(x,y)=x+e^y$ and let $v(x,y)$ be given by the gradient of $f$ norma …
7
votes
Integrability of derivatives
I remember, that there was an example of such a function in the book Counterexamples in Analysis. Just wanted to mention it for the sake of completeness. It can be found in Chapter 8 (Sets and Measure …