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The given function is \( f(x) = \frac{4}{x - 2} - 5x \).

To determine where this function is continuous, we need to check if there are any points where the function is not defined or discontinuous.

### Step 1: Identify points of discontinuity
The function involves a rational expression \( \frac{4}{x - 2} \). This term is undefined when the denominator is zero. So, we solve:
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
At \( x = 2 \), the function is not defined, causing a discontinuity at this point.

### Step 2: Determine continuity elsewhere
The function is composed of:
- A rational term \( \frac{4}{x - 2} \), which is continuous for all \( x \neq 2 \),
- A linear term \( -5x \), which is continuous everywhere.

Thus, the only point of discontinuity is at \( x = 2 \). The function is continuous everywhere else.

### Step 3: Express the set of x-values where the function is continuous using interval notation
Since the function is continuous for all \( x \neq 2 \), the interval of continuity is:
\[
(-\infty, 2) \cup (2, \infty)
\]

Would you like further clarification or more details?

Here are 5 questions for further understanding:
1. Why does the rational function become discontinuous at \( x = 2 \)?
2. How would the graph of \( f(x) \) behave near \( x = 2 \)?
3. Can a function have multiple discontinuities? How would that affect interval notation?
4. What types of functions are continuous everywhere by default?
5. How can limits help in analyzing the continuity of functions?

**Tip:** A function is typically continuous wherever it is defined, but check for discontinuities in cases involving division by zero or undefined points.

At what points is the given function f(x) continuous?
Describe the set of x-values where the function is continuous, using interval notation.
The function is f(x) = 4 / (x - 2) - 5x.

Learn how to determine the continuity of the function f(x) = 4/(x-2) - 5x by identifying points of discontinuity and expressing the intervals of continuity in proper notation. Suitable for students in Grades 10-12.

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