91Ó°ÊÓ

Continuity is an essential property of a function, especially when discussing calculus theorems like the Mean Value Theorem (MVT). A function is continuous on an interval if there are no abrupt jumps, breaks, or holes in its graph over that interval. For the function given in this exercise, \(f(x) = x + |x|\), it is defined piecewise:

• For \(x \geq 0\), \(f(x) = 2x\).
• For \(x < 0\), \(f(x) = 0\).

At \(x = 0\), these two parts of the function connect smoothly as there is no sudden change in the function values. Thus, it is continuous across its domain, \([-2, 1]\).
Continuity allows us to predict the function's behavior, ensuring no unexpected disruptions which are critical in applying various mathematical theorems properly.

Differentiability covers whether a function has a derivative, essentially the slope of the tangent line, at every point within an interval. For a function to be differentiable at a given point, its graph around that point should locally resemble a straight line.
For the function \(f(x) = x + |x|\), it is piecewise, and we assess differentiability as follows:

• For \(x > 0\), the derivative is \(f'(x) = 2\).
• For \(x < 0\), the derivative is \(f'(x) = 0\).

However, at \(x = 0\), the derivatives from the left side and the right side do not coincide: the left-hand derivative is 0, while the right-hand derivative is 2. This discontinuity in slope causes the function to be non-differentiable at \(x = 0\).
Being non-differentiable at a single point in an interval can impede the applicability of the Mean Value Theorem because differentiability is required across an entire open interval.

A piecewise function is a function made up of multiple sub-functions, each applicable to a specific interval. This function, \(f(x) = x + |x|\), is a perfect example of a piecewise function:

• For \(x \geq 0\), it behaves as \(f(x) = 2x\), a line with slope 2.
• For \(x < 0\), it is constant, resembling \(f(x) = 0\), laying flat on the x-axis.

When dealing with piecewise functions, it is crucial to pay attention to how the pieces fit together at their boundaries. In this case, at \(x = 0\), both pieces connect smoothly for continuity but diverge in slope, making the function non-differentiable at that point.
Piecewise functions are common in mathematical modeling where a situation can have different behaviors under different circumstances, thus the importance in understanding their continuity and differentiability.