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The Mean Value Theorem (MVT) is a fundamental concept in calculus that describes a relationship between the average rate of change of a function over an interval and the instantaneous rate of change at a point within that interval. In simpler terms, it guarantees that for a continuous and differentiable function on a closed interval, there exists at least one point in the interval where the function's instantaneous rate of change (its derivative) matches the average rate of change.

However, for the Mean Value Theorem to apply, the function must meet specific criteria: it must be continuous on the closed interval \[a, b\] and it must be differentiable on the open interval \(a, b\). If a function is discontinuous on the interval or not differentiable at any point in the interval, the conditions of the MVT are not satisfied, and we cannot apply the theorem to that function. Understanding when and why the theorem can be applied is crucial for comprehending the behavior of functions within a given range.

In calculus, continuity of a function is a key concept, denoting that the function does not have any abrupt changes, holes, or jumps. Conversely, a discontinuous function exhibits one or more of these characteristics, breaking the flow of the function's graph. Point discontinuity or jump discontinuity are common types, where the function 'jumps' from one value to another abruptly, creating a gap between the segments of the function's graph.

Specifically, as in the step-by-step solution, by defining a function piecewise with separate formulas on different intervals, we can intentionally create discontinuities, such as a jump at x = 0, where the function suddenly shifts from one expression to another. This not only aids in visualizing concepts like continuity, but it also introduces the complications that discontinuities pose for applying certain theorems, such as the Mean Value Theorem.

A piecewise function is a type of function that is defined by different expressions or 'pieces' for various parts of its domain. This allows for more flexibility in describing complex behaviors, where a single formula might not adequately represent the function across its entire range.

To sketch a piecewise function, as in the solution provided, each piece must be considered separately. The domain is partitioned into intervals, and within each interval, the function takes on a specific form. When transitioning from one piece to another, if there's no overlap or smooth continuation, it's important to indicate these changes with open or closed circles to denote discontinuity points. Piecewise functions serve as excellent examples when exploring the ramifications of discontinuity on theorems and properties in calculus.

Graph sketching is a powerful tool in calculus for visualizing the behavior of functions. It assists in understanding limits, derivative concepts, and, as highlighted earlier, the Mean Value Theorem. When sketching graphs, important features to consider include intercepts, peaks and valleys, trends, and of course, points of continuity and discontinuity.

Successfully sketching a graph involves plotting points where the function changes direction (turning points) and where the function is not defined or is discontinuous. In the case of a discontinuous function, we typically depict the location of the discontinuity, as seen in the aforementioned step-by-step solution, with a clear interruption in the line or curve, using an open circle to indicate a point not included in the function or a filled circle to denote a jump. Mastering graph sketching equips students with a visual comprehension of complex calculus concepts, enhancing their analytical skills.