This line of questioning happened today in Calculus class. The actual question was "Would it be possible for a function to have both a sharp point and a smooth part?" I should have asked the student to be more precise and reword his question. But I knew the question he was really asking was: "Is it possible for a function to have both a relative max. or min. with an undefined derivative and a relative max. or min. with a defined derivative?" Although I could have (and maybe should have) thrown the question back to my entire class, in a matter of about 15 seconds, I produced the following graph in Desmos and I asked "Why do we have cusps on this graph?" The student recognized that it was due to the absolute value being used in the function.
Next, the student asked if it would be possible to have a cusp in the middle of the graph. Within another few seconds, I produced the following graph. To which I heard a student whisper, "that's cool". Thanks again to Desmos for making it easier for me to keep me and my students curious.