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Найти точки разрыва функции
f
(
x
)
и определить их характер.

To find the discontinuities of the function \( f(x) \) and determine their nature, let's analyze the piecewise function given:

\[
f(x) = \begin{cases} 
e^{\frac{1}{\operatorname{arctg} \, x}}, & x < 0 \\ 
\frac{\sqrt{1 - \sin \frac{\pi^2}{2x}}}{\pi - x}, & x > 0 
\end{cases}
\]

Here's the approach we'll take to solve this problem:

1. **Identify Potential Points of Discontinuity:**
   Since this is a piecewise function with different definitions for \( x < 0 \) and \( x > 0 \), we should examine the behavior of \( f(x) \) at \( x = 0 \) (the boundary between the two cases) as it’s a possible point of discontinuity.

2. **Examine Left and Right Limits at \( x = 0 \):**
   - **Left-hand limit (\( x \to 0^- \)):** Evaluate \( \lim_{x \to 0^-} e^{\frac{1}{\operatorname{arctg} \, x}} \).
     - As \( x \to 0^- \), \( \operatorname{arctg} \, x \to 0 \), so \( \frac{1}{\operatorname{arctg} \, x} \to \infty \).
     - Consequently, \( e^{\frac{1}{\operatorname{arctg} \, x}} \to e^{\infty} = \infty \).
     - This means the left-hand limit does not approach a finite value; instead, it tends to infinity.
   
   - **Right-hand limit (\( x \to 0^+ \)):** Evaluate \( \lim_{x \to 0^+} \frac{\sqrt{1 - \sin \frac{\pi^2}{2x}}}{\pi - x} \).
     - The behavior of \( \sin \frac{\pi^2}{2x} \) as \( x \to 0^+ \) is oscillatory, leading to undefined or oscillating behavior in the square root and denominator.
     - This makes it difficult to approach a single finite limit from the right side as \( x \to 0^+ \).

3. **Conclusion on Discontinuity at \( x = 0 \):**
   Since both the left-hand and right-hand limits are not finite and do not match, there is a **discontinuity at \( x = 0 \)**. Specifically, this discontinuity is an **essential (infinite) discontinuity** because the limits do not converge to any finite value.

In summary, the function \( f(x) \) has a discontinuity at \( x = 0 \), and it is an essential discontinuity due to divergent behavior from both sides.

Would you like further details on any step or a deeper exploration of the function's properties?

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### Related Questions:
1. What is an essential discontinuity and how does it differ from a removable or jump discontinuity?
2. How does the oscillatory nature of \( \sin \frac{\pi^2}{2x} \) affect the limit as \( x \to 0^+ \)?
3. Could we use series expansions to approximate the behavior of \( f(x) \) around \( x = 0 \)?
4. How would the graph of this function look near \( x = 0 \)?
5. What would happen if we defined \( f(0) \) in some way to attempt to remove the discontinuity?

### Tip:
When working with piecewise functions, always check boundary points between cases, as these are often where discontinuities occur.

Найти точки разрыва функции f(x) и определить их характер.

Analyze the discontinuities of the piecewise function f(x) with definitions for x < 0 and x > 0. Learn how to examine the left and right limits at x = 0 to determine the type of discontinuity, and explore oscillatory behavior and infinite discontinuities.

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