Kurs:Mathematik für Anwender (Osnabrück 2019-2020)/Teil I/Repetitorium/19/stetig nicht differenzierbar/Studentenfrage/Antwort

I'm not sure what you mean with period. To me a period in mathematics is one of the following: w:en:Periodic_function or w:en:Period_(algebraic_geometry). I'm not sure how either is applicable here, but from context I assume you mean an interval.

Yes, a function can be continuous over an interval ${}I$ and at the same time not differentiable over the same interval ${}I$.

Take for example

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto \vert {x}\vert ,$

i.e. the absolute value function which was the topic of Aufgabe 10.3 and Aufgabe 14.3. It is continuous (you can use ${}\delta =\epsilon$ in the definition of continuity). You can show that it is not differentiable at ${}0$ by comparing difference quotients approaching ${}0$ from the left and from the right. From the left the differential would have to be -1 and from the right it would have to be 1.

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