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
and at the same time not differentiable over the same interval
.
Take for example
-
i.e. the absolute value function which was the topic of
Aufgabe 10.3 and
Aufgabe 14.3. It is continuous (you can use
![{\displaystyle {}\delta =\epsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c70dc2822c549bb8ce0685e57158354124b0157)
in the definition of continuity). You can show that it is not differentiable at
![{\displaystyle {}0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5428e3b06006771c083bd17ed8fce8f3be334b2)
by comparing difference quotients approaching
![{\displaystyle {}0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5428e3b06006771c083bd17ed8fce8f3be334b2)
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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