The problem asks:

Suppose the function $f$ is differentiable at all points of the real line. Does it follow that $|f|$ must also be differentiable at all points of the real line? If your answer is yes, explain why this must be true. If your answer is no, provide an example of a function $f$ that is differentiable at all points of the real line, such that $|f|$ is not differentiable at least at one point.

Answer:

No, it does not follow that $|f|$ must also be differentiable at all points of the real line, even if $f$ is differentiable everywhere. The absolute value function $|f|$ can fail to be differentiable at points where $f(x) = 0$, even if $f$ itself is differentiable at that point.

Example:

Consider the function $f(x) = x$. This function is differentiable everywhere on the real line. However, consider its absolute value $|f(x)| = |x|$. The function $|x|$ is not differentiable at $x = 0$, even though $f(x) = x$ is differentiable at $x = 0$.

• For $x > 0$, $|x| = x$, so the derivative is $1$.
• For $x < 0$, $|x| = -x$, so the derivative is $-1$.

At $x = 0$, the left-hand derivative (coming from the negative side) is $-1$, and the right-hand derivative (coming from the positive side) is $1$. Since these two values are not equal, $|x|$ is not differentiable at $x = 0$.

Thus, we have an example where $f(x) = x$ is differentiable everywhere, but $|f(x)| = |x|$ is not differentiable at $x = 0$.

Would you like more details on this concept or further examples?

Here are some related questions to deepen understanding:

1. What conditions must be met for $|f|$ to be differentiable at a point where $f(x) = 0$?
2. Can a function $f$ be differentiable at a point where its absolute value $|f|$ is not?
3. How does the behavior of a function around $x = 0$ impact the differentiability of $|f|$?
4. What is the geometric interpretation of a function failing to be differentiable at a point where it crosses the x-axis?
5. What is the relationship between continuity and differentiability for the function $|f(x)|$?

Tip: Always check the behavior of both the function and its derivative around critical points (like zeros of the function) to analyze differentiability issues.