https://file.aiquickdraw.com/mathbot/file/1727309178201rsby6as1.jpg

Let's analyze the piecewise function:

\[
f(x) =
\begin{cases}
3x, & \text{if } x < 0 \\
2x^2, & \text{if } 0 \leq x \leq 5 \\
2x + 3, & \text{if } x > 5
\end{cases}
\]

We are asked to determine which of the following is true regarding the continuity and discontinuities of the function. Let's analyze each part:

### 1. **At \(x = 0\):**

- Left-hand limit (\(x \to 0^-\)):  
  Using the first piece \(3x\),  
  \[
  \lim_{x \to 0^-} f(x) = 3(0) = 0
  \]
  
- Right-hand limit (\(x \to 0^+\)):  
  Using the second piece \(2x^2\),  
  \[
  \lim_{x \to 0^+} f(x) = 2(0)^2 = 0
  \]
  
- Value of \(f(0)\):  
  Since \(f(x) = 2x^2\) when \(x = 0\),  
  \[
  f(0) = 2(0)^2 = 0
  \]

Since the left-hand limit, right-hand limit, and the function value at \(x = 0\) all match, **the function is continuous at \(x = 0\)**.

### 2. **At \(x = 5\):**

- Left-hand limit (\(x \to 5^-\)):  
  Using the second piece \(2x^2\),  
  \[
  \lim_{x \to 5^-} f(x) = 2(5)^2 = 50
  \]
  
- Right-hand limit (\(x \to 5^+\)):  
  Using the third piece \(2x + 3\),  
  \[
  \lim_{x \to 5^+} f(x) = 2(5) + 3 = 13
  \]

Since the left-hand limit (\(50\)) and the right-hand limit (\(13\)) are not equal, **there is a jump discontinuity at \(x = 5\)**.

### Conclusion:
- The function is continuous at \(x = 0\).
- The function has a jump discontinuity at \(x = 5\).

### Correct Answer:
The first option:  
- "This function is continuous at \(x = 0\), it has a jump discontinuity at \(x = 5\)."

Would you like further clarification or more details? Here are 5 related questions to expand on this topic:

1. How do you identify whether a function has a jump discontinuity?
2. What are the different types of discontinuities in functions?
3. What is the importance of left-hand and right-hand limits in checking continuity?
4. Can a function have both removable and jump discontinuities at the same point?
5. How does the definition of a piecewise function affect its continuity properties?

**Tip:** Always check both the limits and the actual value of the function at the points of interest when determining continuity.

Consider the following piecewise function: 
 f(x) = 
 { 3x, if x < 0 
 2x^2, if 0 < x ≤ 5 
 2x + 3, if 5 < x } 
 Which of the following is true about this function?

Learn how to determine the continuity and discontinuities of a piecewise function. This problem involves analyzing left-hand and right-hand limits and identifying jump discontinuities. Suitable for Grade 12 or College Level students.

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