Problem 4. Define the function f : R → R by f (x) = max{0, x}. For each a ∈ R, determineif f is differentiable at a and prove your answer

The function f(x) = max{0, x} is also known as the ReLU (Rectified Linear Unit) function in machine learning. It is defined as f(x) = 0 for x < 0 and f(x) = x for x >= 0.

To determine if f is differentiable at a, we need to check if the derivative exists at that point.

1. For a < 0, f(x) = 0, which is a constant function. The derivative of a constant function is 0, so f is differentiable for a < 0.

2. For a > 0, f(x) = x, which is a linear function. The derivative of a linear function is the slope of the line, which is 1 in this case. So, f is differentiable for a > 0.

3. The point a = 0 needs special consideration. The right-hand limit of the derivative as x approaches 0 is 1 (from the x part of the function), and the left-hand limit is 0 (from the 0 part of the function). Since the right-hand limit and the left-hand limit are not equal, the derivative at x = 0 does not exist.

Therefore, the function f(x) = max{0, x} is differentiable for all real numbers except at x = 0.