The document appears to be an assignment related to calculus, particularly focusing on a function $f(x)$ and its derivative. Here's a breakdown of the problem as described:

You are asked to analyze the behavior of a function $f$ on a given interval, excluding $x = 0$, where the derivative exists everywhere except at that point. Specifically, the problem inquires about how many values of $x$ satisfy the given condition for the function in the interval $a \leq x \leq b$.

There is also a limit condition to evaluate: $\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = \frac{f(b) - f(a)}{b - a}$ This resembles the definition of the derivative using limits, suggesting that you need to analyze the slope or tangent of the function at certain points.

To proceed, you will likely need to determine the critical points of $f(x)$ where its derivative is zero or undefined (except at $x = 0$), and check the behavior of the function within the interval.

Would you like to explore how to calculate these values or solve this limit problem in detail? Here are some related questions that could help expand your understanding:

1. How do we find critical points of a function on a given interval?
2. How does the derivative behave near $x = 0$, and why might it be undefined there?
3. What role does the Mean Value Theorem play in problems like this?
4. How do you apply limits to find the slope of a function between two points?
5. What types of discontinuities can exist in the derivative of a function?

Tip: When working with limits involving the derivative, visualize the slope of the secant line connecting two points as $x$ approaches $c$ to understand the behavior of the function better.