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Dec 13, 2022 at 0:56 vote accept Iosif Pinelis
Dec 11, 2022 at 16:30 comment added Iosif Pinelis Great! I will wait a bit for other answers. If no more answers are given, I am going to accept yours.
Dec 11, 2022 at 16:26 comment added Kostya_I @IosifPinelis indeed - corrected.
Dec 11, 2022 at 16:24 history edited Kostya_I CC BY-SA 4.0
small mistake corrected
Dec 11, 2022 at 13:45 comment added Iosif Pinelis Thank you for your answer. Concerning your second example: actually, $\partial_t f(t,0)|_{t=0}=\infty\ne0$. I had thought about an example like this but did not finish that idea, ending up with that different one. I suspect one can modify the rescaled-normal-pdf idea to get $\partial_t f(t,x)|_{t=0}\equiv 0$ indeed.
Dec 11, 2022 at 12:39 history edited Kostya_I CC BY-SA 4.0
added 798 characters in body
Dec 11, 2022 at 10:54 history answered Kostya_I CC BY-SA 4.0