Let's analyze the function you provided:

$f(x) = \frac{x}{100 - 2x} + 37 + 75$

This function is composed of two main parts:

1. $\frac{x}{100 - 2x}$ (a rational function)
2. The constants $37 + 75 = 112$

Step-by-step analysis:

1. Continuity of the rational function $\frac{x}{100 - 2x}$:

A rational function is continuous wherever its denominator is not equal to zero. Here, the denominator is $100 - 2x$, so we need to find where it is zero:

$100 - 2x = 0 \quad \Rightarrow \quad x = \frac{100}{2} = 50$

Thus, the function is not continuous at $x = 50$ because the denominator becomes zero, causing a division by zero.

2. The constants $37 + 75 = 112$:

Constants are continuous everywhere.

Conclusion:

The overall function $f(x) = \frac{x}{100 - 2x} + 112$ is continuous everywhere except at $x = 50$, where the rational part of the function becomes undefined.

Continuity Interval:

The function is continuous on the intervals:

$(-\infty, 50) \cup (50, \infty)$

Would you like more details or have any questions? Here are some related problems you might consider:

1. What is the behavior of $f(x)$ as $x \to 50$?
2. How would you determine the limits of $f(x)$ at the endpoints of the intervals of continuity?
3. What happens to the value of $f(x)$ as $x \to \infty$ or $x \to -\infty$?
4. How would the intervals change if the denominator were $100 - x$ instead of $100 - 2x$?
5. Can we modify the function to make it continuous at $x = 50$?

Tip: Rational functions are continuous everywhere their denominators are non-zero. Keep an eye on points where the denominator can become zero.