The following two paragraphs are the last footnote on p. 69 of . I found this such good advice that I began the first chapter of my 1993 Ph.D. dissertation, on p. 6, with this quote.
 William Henry Young, On the distinction of right and left at points of discontinuity, Quarterly Journal of Pure and Applied Mathematics 39 (1908), pp. 67−83. (Also here.)
Mark the importance of testing not only the accuracy but also the scope of one's results by constructing examples. To quote an instance which has come under my notice in the course of my present work, Dini (p. 307) states that if a left-hand derivate and a right-hand derivate both exist and are finite and different at every point of an interval $\ldots$ $\ldots$ certain results follow.
The reader might well imagine not only that such a case could occur, but that Dini knew of a case where it did occur. As a matter of fact, however, the hypthesis [sic] is an impossible one. In default of an example it could, in such a case, only stimulate research to state that an example had not been found.
Incidentally, I don’t know whether “p. 307” is for the 1878 Italian original of his real functions book or for the 1892 German translation of his real functions book. Young’s previous footnote appears to cite the 1878 Italian original, but p. 307 of the German translation seems more likely (based on math symbols appearing; I can’t read German or Italian).
For some more context about the fact that no such function exists, see B. S. Thomson’s answer to If $f$ is bounded and left-continuous, can $f$ be nowhere continuous? and my answers to A search for theorems which appear to have very few, if any hypotheses and Real-valued function of one variable which is continuous on $[a,b]$ and semi-differentiable on $[a,b)$?