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### Problem Summary:
You are asked to:
- **Determine the points** on the interval \( [0,4] \) where the function has a discontinuity.
- For each discontinuity point, **state which condition** from the continuity checklist is violated:
  1. \( f(a) \) is defined (i.e., \( a \) is in the domain of \( f \)).
  2. \( \lim_{x \to a} f(x) \) exists.
  3. \( \lim_{x \to a} f(x) = f(a) \) (i.e., the value of the limit equals the function's value).

The graph represents the function \( f(x) \) over the interval \( [0,4] \).

### Analysis of the Graph:
- The graph is continuous except at **\( x = 2 \)**, where there is a **jump discontinuity**.
  - At \( x = 2 \), the left-hand limit \( \lim_{x \to 2^-} f(x) \) is not equal to the right-hand limit \( \lim_{x \to 2^+} f(x) \).
  - Therefore, the function \( f(x) \) is discontinuous at \( x = 2 \).

### Violation of Continuity Conditions at \( x = 2 \):
- \( f(2) \) is defined, so the first condition holds.
- However, \( \lim_{x \to 2} f(x) \) does not exist because the left-hand limit and right-hand limit differ.
- Therefore, the second condition is violated.

### Answer:
- The function is **discontinuous at \( x = 2 \)**.
- The violated condition is: **the limit does not exist** (condition 2).

Would you like any more details on this? Here are some related questions that can expand your understanding:

1. What are the different types of discontinuities (jump, removable, infinite)?
2. How do you calculate limits graphically?
3. Why does a discontinuity at \( x = 2 \) in this graph count as a jump discontinuity?
4. What is the difference between left-hand and right-hand limits?
5. Can a function have more than one discontinuity within a given interval?

**Tip:** Always check both one-sided limits when looking for discontinuities in graphs!

Determine the points on the interval [0,4] where the function has a discontinuity. At each point, state the violated conditions of continuity.

This problem involves identifying the points of discontinuity in a function on the interval [0,4] and checking the violated continuity conditions, such as whether the function is defined, the limit exists, or the limit equals the function's value.

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