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The function \( f(x) \) is defined as a piecewise function with two cases:

\[
f(x) = 
\begin{cases} 
\arctan \left( \frac{1}{x} \right), & x < 1 \\ 
\frac{\pi x}{4}, & x \geq 1 
\end{cases}
\]

To analyze this function, we may want to:
1. Check for continuity at \( x = 1 \).
2. Determine the differentiability at \( x = 1 \).
3. Explore the behavior of each piece of the function for values approaching \( x = 1 \).
4. Analyze the range of \( f(x) \).
5. Calculate specific values of \( f(x) \) for particular \( x \)-values, like \( x = 0.5 \) and \( x = 2 \).

Would you like to go through any of these details, or do you have a specific question about this function?

Here are 5 additional related questions for further exploration:
1. What is the limit of \( f(x) \) as \( x \) approaches 1 from the left and right?
2. Is \( f(x) \) continuous at \( x = 1 \)?
3. Can we find the derivative of \( f(x) \) in each piece and determine if \( f(x) \) is differentiable at \( x = 1 \)?
4. How does the behavior of \( \arctan \left( \frac{1}{x} \right) \) as \( x \to 1^- \) compare to \( \frac{\pi x}{4} \) as \( x \to 1^+ \)?
5. What is the range of \( f(x) \) over its entire domain?

**Tip:** To check continuity at a specific point, ensure that the left-hand limit and right-hand limit at that point match the function's value at that point.

Determine properties of the piecewise function f(x) = { arctg(1/x), x < 1; πx/4, x ≥ 1 }.

Explore the continuity and differentiability of a piecewise function f(x) = { arctg(1/x), x < 1; πx/4, x ≥ 1 } at x = 1. Learn key mathematical concepts such as limits, continuity, and differentiability, suitable for high school students.

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