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 Apr 26 awarded Peer Pressure Apr 26 comment Weird subfields of $\Bbb{R}$ @BeniBogosel Ok, thanks anyway for taking the time to answer after all this time. Apr 24 awarded Tumbleweed Apr 21 comment Weird subfields of $\Bbb{R}$ @BeniBogosel How have you "found" this question ? Did you come up with it on your own or have you found it in a book/article etc. ? (Sorry for the late commentary on a question know 5 years old.) Apr 17 accepted Continuous $(x,y) \to k(x,y)$ with “discontinous slope-behavior” at one $x\to k(x,y)$-slice: Is this possible? Apr 17 comment Discontinuous derivative *not* by oscillation @HansLundmark You'd make me happy if you could you explain that a little bit further. If possible, maybe indicate an example of two such discontinuity-inducing oscillations which one would classifiy as being "distinct". Apr 17 comment Discontinuous derivative *not* by oscillation @MartinSleziak Besides splitting the jump discontinuity case into two, that link doesn't explain what kind of behavior can occur in the oscillating case. Apr 17 accepted Discontinuous derivative *not* by oscillation Apr 17 comment Continuous $(x,y) \to k(x,y)$ with “discontinous slope-behavior” at one $x\to k(x,y)$-slice: Is this possible? If you post that as an answer I'll accept it. Otherwise I'll delete the question. Apr 17 comment Continuous $(x,y) \to k(x,y)$ with “discontinous slope-behavior” at one $x\to k(x,y)$-slice: Is this possible? Won't work. Assuming you define this on, say, $[0,1]\times [\varepsilon,1]$ for some small $\varepsilon >0$ then the slope is everywhere finite. If you define it on $[0,1]\times [0,1]$ then $k$ is not continuous in $(0,0)$ anymore (actually, its not even defined there, but also can't be continuously extended to that point). Please check your suggestions carefully, finding such an example does not seem to be that trivial. Apr 17 asked Criteria for Lipschitz continuity in case the function is not assumed to be (everywhere) differentiable. Apr 17 comment Discontinuous derivative *not* by oscillation My first question wasn't actually precise: With "other" I mean "other then oscillatory discontinuity and the jump discontinuity you mentioned - in which case I assume the answer would be "No", right ? Apr 17 comment Discontinuous derivative *not* by oscillation So I take your "Yes" pertains to the second question ? In that case I find the wording (including the link you provided) not really good (but probably one can't do anything about that), because if the discontinuity of second kind is "everything that is not a jump discontinuity" this somehow implies that there are distinct types of behavior for second kind discontinuities. But then proceeding calling it "oscillatory discontinuity" actually reveals that there is only one type of behavior Apr 17 comment Discontinuous derivative *not* by oscillation Are the other types of discontinuities in the one-variable case ? (I think not, but I'm not sure...) Does that imply that in the one-variable case oscillatory behavior is the only way to force the derivative to become discontinuous ? Apr 17 asked Continuous $(x,y) \to k(x,y)$ with “discontinous slope-behavior” at one $x\to k(x,y)$-slice: Is this possible? Apr 17 asked Discontinuous derivative *not* by oscillation Mar 30 awarded Yearling Feb 29 accepted Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up Feb 28 comment Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up Second: Wow, these are some amazing reasons to study complex numbers/analysis. I actually can't remember when was the last time mathematical connection thrilled me in such a way! (May I respectfully ask if you perhaps know of any similar *amazing*/compelling reason to specifically study complex Hilbert spaces ?) Feb 28 comment Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up +1 First: I must confess it feels good when a professor agrees that textbooks provide bad motivation - and yes what you write makes perfect sense! (Do you know of any introductory textbooks - say at the level of a beginning graduate student, no matter which subject, I just like to know of them for future reading - that do not do a lousy job regarding motivation ? Where the ordering of motivating examples and theorems/definitions isn't inverted ?)