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A function is said to be differentiable at a particular point if it has a defined tangent at that point, which means we can calculate its derivative there. In simpler terms, if you can "draw" the function's curve at that point smoothly without picking up your pencil, then it is differentiable.
For the function in question, which is defined as a series sum of convex functions, it's crucial that this series behaves well outside a particular set. Specifically, the series is differentiable on the set of all real numbers excluding the set \( A \), denoted as \( \mathbb{R} \setminus A \).
Each term in the series, \( \frac{\|x-a_k\|}{10^k} \), is differentiable because the absolute value function, while not differentiable at zero, is differentiable everywhere else.
The derivative of each term contributes to the derivative of the sum, making \( f \) differentiable wherever \( x eq a_k \). The series of derivatives itself converges due to the exponential decay factor \( 1/10^k \). This ensures that the function behaves nicely and has a well-defined slope almost everywhere.

Sometimes, a function can be non-differentiable at certain points. This often happens at points where the function has sharp turns or "corners". In our case, the function \( f \) is non-differentiable at every point \( a_k \) belonging to the set \( A \).
Why does this happen? At any one of these points, the expression \( \frac{\|x-a_k\|}{10^k} \) features a corner because the absolute value of a number changes slope abruptly when crossing the point it's centered on.
This means that the derivative approaches different values from different sides; hence, the slope isn't single-valued. Even when summed with other similar terms, the discontinuity at \( a_k \), inherent in each term, ensures that \( f \) remains non-differentiable at these specific points.

Understanding series convergence is about determining if adding an infinite number of terms gives us a finite result. For the function \( f \), which is a series sum, convergence is pivotal for ensuring the function is well-defined and behaves predictably.
The series \( \sum_{k=1}^{\infty} \frac{\|x-a_k\|}{10^k} \) demonstrates convergence due to the factors \( 1/10^k \), which decay exponentially as \( k \) increases.

• Exponential decay makes the series' terms get smaller quickly.
• Smaller terms ensure that the entire sum reaches a limit, preventing it from blowing up to infinity.

This convergence is essential for differentiability; the uniform convergence of both the function and its derivative ensures our results hold true over the real numbers, excluding \( A \). This means we can rely on the sum not only to converge but to do so in a manner that supports differentiability where expected.

In mathematics, countable sets are those that can be matched one-to-one with the set of natural numbers. They can be finite or infinite, but crucially, they're enumerative.
The set \( A \), comprising all the points \( a_k \), is countable, meaning it can be listed as \( \{a_1, a_2, a_3, \ldots\} \). Think of countability as being able to 'count' the elements, even if the counting never ends.

• Countable sets can theoretically include infinitely many elements, like all integers.
• This property is contrasted with uncountable sets, like the real numbers between any two given points, which cannot be matched to a simple list.

The countability of \( A \) implies it doesn’t take up "space" in the same way as the rest of the real numbers. Hence, the points where \( f \) is non-differentiable, though infinite, are "thinly" spread, allowing \( f \) to be differentiable wherever \( x \) is not in \( A \).