An example of a function $f(x)$ that satisfies the following conditions:

• Domain: $(-\infty, 0) \cup (0, \infty)$
• Derivative: $f'(x) = 0$
• Function: $f(x)$ is not constant

is:

$f(x) = \frac{1}{x}$

Explanation:

• Domain: The function $f(x) = \frac{1}{x}$ is defined for all $x \neq 0$, so its domain is $(-\infty, 0) \cup (0, \infty)$.

• Derivative: The derivative of $f(x)$ is computed as follows:

  $f'(x) = \frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}$

  Notice that $f'(x) = -\frac{1}{x^2}$ is never zero for any $x$ in the domain of $f(x)$. Therefore, we need to rethink the condition.

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Here’s the correct example:

Consider the function $f(x) = \text{sgn}(x)$ where $\text{sgn}(x)$ is the sign function:

1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \end{cases}$$ - **Domain**: The function is defined for all $$x \neq 0$$, so its domain is $$(-\infty, 0) \cup (0, \infty)$$. - **Derivative**: $$f(x)$$ is constant for both $$x > 0$$ and $$x < 0$$, hence its derivative in both regions is $$0$$. But because of the jump discontinuity at $$x = 0$$, the function is not considered continuous and, thus, **not a constant function** overall. Would you like to explore more examples or the reasoning in more detail? --- ### Here are 5 related questions for deeper exploration: 1. How does the derivative of piecewise functions behave near discontinuities? 2. What are some other common functions with jump discontinuities? 3. How can we classify functions that have a derivative of zero but are not constant? 4. Can functions with removable discontinuities satisfy $$f'(x) = 0$$ at all points? 5. How does the behavior of the function around critical points affect its overall nature? ### Tip: When dealing with piecewise functions, always check for continuity and differentiability in each region separately to understand their behavior fully.