Let's break down the problem from the image you uploaded.

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!