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Sketching a graph of a piecewise function like \( f(x) \) involves graphing each individual piece separately and then combining them. To start, we need to understand each piece of the function. For \( x \leq 0 \), the function is \( -x \), which is linear with a negative slope of -1, meaning it descends through the origin. This results in a straight line sloping downwards to the left. For \( x > 0 \), the function is a constant \( 0 \), appearing as a horizontal line on the x-axis. It's critical to note what happens at \( x = 0 \). Here, the graph jumps, indicating a change from the line \( -x \) to the line at \( y = 0 \). This distinct change, or jump, helps in identifying any points of discontinuity or nondifferentiability, which is a key feature in sketching piecewise graphs. When sketching, it's important to plot clearly each segment according to its definition and to mark the transition points effectively.

The concept of differentiability refers to whether the derivative of a function exists at a particular point. In simpler terms, it tells us if there's a defined, tangent slope at that point. For the function \( f(x) = -x \) for \( x \leq 0 \), it is a straight line, hence differentiable everywhere in its domain, with the derivative \( f'(x) = -1 \). For the section \( x > 0 \) where \( f(x) = 0 \), it's a constant function, meaning its derivative is zero, so \( f'(x) = 0 \). Differentiability depends heavily on both the continuity and smoothness of the function. At any point where the function fails to be smooth, such as sharp turns or jumps, differentiability ceases to exist for those points. Especially for piecewise functions, checking differentiability involves ensuring that both continuity and the behavior of the function around the transition points are consistent.

Discontinuity occurs when a function has sudden jumps or breaks in its graph. In our function \( f(x) \), the discontinuity can clearly be seen at \( x = 0 \). Here, the function jumps from \( -x \) to \( 0 \), making the left-hand limit \( f(x) = 0 \) and the right-hand limit \( f(x) = -0 \) unequal. These differing limits indicate a sudden break, leading to discontinuity. Discontinuities are crucial in graph analysis since they often signal areas where the derivative can't be calculated, pointing to weakness or sharpness in the graph. Such breaks can occur in piecewise functions, emphasizing the importance of checking the transition points between segments. Recognizing points of discontinuity provides insight into the structure and behavior of the graph, influencing the function's differentiability and overall continuity.

Derivatives provide us with the rate of change of a function, essentially the slope of the tangent at any given point on its graph. For the piecewise function \( f(x) \), we determine its derivative, \( f'(x) \), separately across its defined intervals. In the part where \( f(x) = -x \) and \( x \leq 0 \), the function behaves linearly, leading to a constant derivative \( f'(x) = -1 \), indicating a downward slope. On the other hand, for \( x > 0 \) where we're dealing with a constant function \( f(x) = 0 \), the derivative remains \( f'(x) = 0 \), confirming there is no slope as the line is flat. Derivatives illustrate not only continuous change but also highlight points where a function might lack such smooth transitions. This is essential when graphing as it points out any sharp changes, necessary for identifying nondifferentiable points.