To determine if the derivative of the function $f(x) = \frac{x}{\lfloor x \rfloor + 1}$ is continuous, let's analyze it step by step:

1. The Function $f(x)$

The function $f(x) = \frac{x}{\lfloor x \rfloor + 1}$:

• $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
• The value of $\lfloor x \rfloor$ changes discontinuously at integer values of $x$.

Because $\lfloor x \rfloor$ is piecewise constant, $f(x)$ is piecewise defined over intervals of the form $[n, n+1)$ for integer $n$. Within each interval: $f(x) = \frac{x}{n+1}, \text{ where } n = \lfloor x \rfloor.$

2. Differentiability of $f(x)$

Within each interval $[n, n+1)$, $\lfloor x \rfloor$ is constant. The derivative is: $f'(x) = \frac{d}{dx} \left( \frac{x}{n+1} \right) = \frac{1}{n+1}.$ Thus, $f(x)$ is differentiable within each interval $[n, n+1)$.

At the boundaries ($x = n$, where $n$ is an integer), the function $f(x)$ can have a discontinuity in its derivative because $\lfloor x \rfloor$ jumps.

3. Continuity of the Derivative

At $x = n$ (integer points):

• As $x$ approaches $n$ from the left ($x \to n^-$), the derivative is $\frac{1}{n+1}$.
• As $x$ approaches $n$ from the right ($x \to n^+$), the derivative is $\frac{1}{n+2}$.

These two values of the derivative do not match at $x = n$, meaning the derivative is discontinuous at integer values of $x$.

Conclusion

The derivative of $f(x) = \frac{x}{\lfloor x \rfloor + 1}$ is not continuous. The points of discontinuity occur at all integers $x$.

Would you like a graph of $f(x)$ and its derivative for better visualization?

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Expanding Thoughts

Here are 5 related questions:

1. What happens to the continuity of $f(x)$ itself at integer points?
2. How does the choice of denominator ($\lfloor x \rfloor + 1$) affect the behavior of $f(x)$?
3. Can the derivative of a piecewise-defined function ever be continuous across the boundaries?
4. What would happen if the denominator was $\lfloor x \rfloor - 1$ instead of $\lfloor x \rfloor + 1$?
5. How does the floor function affect differentiability compared to the ceiling function?

Tip

When analyzing derivatives of functions involving the floor or ceiling function, always check the behavior at the discontinuities of the floor/ceiling function explicitly!