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onku kyseessä jatkuva vai hyppäysepäjatkuvuus, poistuva epäjatkuvuus vai epäjatkuvuus äärettömän takia.

The given function \( f(x) \) is defined piecewise as:

\[
f(x) = 
\begin{cases} 
\frac{1}{x + 1}, & \text{kun } x \geq 1 \\
\frac{1}{x - 3} + 1, & \text{kun } x < 1 
\end{cases}
\]

To determine the type of discontinuity, let's analyze the continuity at the critical point \( x = 1 \).

### 1. Check Continuity at \( x = 1 \)

**Left-hand limit (as \( x \to 1^- \)):**
\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} \left(\frac{1}{x - 3} + 1\right) = \frac{1}{1 - 3} + 1 = -\frac{1}{2} + 1 = \frac{1}{2}.
\]

**Right-hand limit (as \( x \to 1^+ \)):**
\[
\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} \frac{1}{x + 1} = \frac{1}{1 + 1} = \frac{1}{2}.
\]

**Function value at \( x = 1 \):**
\[
f(1) = \frac{1}{1 + 1} = \frac{1}{2}.
\]

### 2. Conclusion

Since the left-hand limit, right-hand limit, and the function value at \( x = 1 \) are all equal to \( \frac{1}{2} \), the function is continuous at this point. There is no jump, removable, or infinite discontinuity.

### Answer:
**The function is continuous at \( x = 1 \).**

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Do you want more details or have any questions?

### Further Questions:
1. What are the types of discontinuities in general and their characteristics?
2. How would the analysis change if one of the limits differed from the function value?
3. Can you provide examples of jump and removable discontinuities?
4. What are some techniques to check the continuity of more complex piecewise functions?
5. How does continuity affect the integrability of functions?

**Tip:** Continuity is crucial for many calculus operations like integration and differentiation; always check limits from both sides!

Onko kyseessä jatkuva vai hyppäysepäjatkuvuus, poistuva epäjatkuvuus vai epäjatkuvuus äärettömän takia?

This page provides a detailed analysis of continuity and types of discontinuities (jump, removable, or infinite) in a piecewise function. The function f(x) is evaluated at critical points using limits, ensuring that all conditions for continuity are checked. Suitable for high school students in grades 10-12.

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