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text[:1000]The uploaded PDF appears to contain mathematical exercises, likely related to analysis, including topics such as continuity, differentiability, and derivatives. Here is a summary of the first few exercises:

1. Exercise 1:
   It asks whether certain functions are continuous and differentiable over specific intervals. The functions are defined piecewise and involve expressions like $x \sin\left(\frac{1}{x}\right)$ and $x^2 \sin\left(\frac{1}{x}\right)$.

2. Exercise 2:
   This involves calculating the derivatives of the following functions:

   • $f_1(x) = \arctan(e^x)$
   • $f_2(x) = \arctan(\sinh(x))$
   • $f_3(x) = \arctan(\tanh(x^2))$

3. Exercise 3:
   Involves determining under what conditions a function defined piecewise is differentiable. The function involves compositions of $f$ with scaled arguments.

4. Exercise 4:
   Focuses on proving a limit related to the definition of the derivative at a point and asking about the differentiability of a function at a specific point.

Would you like assistance with solving any of these exercises or more details on a specific one?