Math Problem Statement
onku kyseessä jatkuva vai hyppäysepäjatkuvuus, poistuva epäjatkuvuus vai epäjatkuvuus äärettömän takia.
Onko kyseessä jatkuva vai hyppäysepäjatkuvuus, poistuva epäjatkuvuus vai epäjatkuvuus äärettömän takia?
Solution
The given function is defined piecewise as:
\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$$.** --- 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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Discontinuity
Formulas
Left-hand limit: \lim_{x \to a^-} f(x)
Right-hand limit: \lim_{x \to a^+} f(x)
Continuity condition: \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)
Theorems
Limit Theorem
Continuity Theorem
Suitable Grade Level
Grades 10-12