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Calculus is a vast field of mathematics that deals with the study of rates of change (differential calculus) and accumulation of quantities (integral calculus). At the core of calculus is the concept of limits, which provides a rigorous way to approach the idea of instantaneous speed or area under the curve. Calculus allows us to understand the behavior of functions and systems in a dynamic and precise way, enabling us to solve complex problems in engineering, physics, economics, and more.

One of the fundamental theorems in calculus is the Mean Value Theorem. It connects the concept of derivatives (rates of change) with the difference in function values over a certain interval, guaranteeing a particular concordance between the average rate of change and the instantaneous rate of change at least at one point in that interval. It requires a function to be continuous over the closed interval and differentiable over the open interval, which are conditions rooted deeply in the nature of calculus.

The average rate of change is a concept that describes how much a quantity changes on average over a particular interval. In the context of functions, it is the change in the function's value divided by the change in the input value over an interval. Mathematically, if we have a function f(x) defined on an interval [a, b], the average rate of change is expressed as \( \frac{f(b)-f(a)}{b-a} \).

This concept is analogous to the average speed of an object moving along a path; it doesn't tell us about the object's speed at a specific point but gives a general idea of the object's speed over the distance traveled. In calculus, understanding the average rate of change sets the stage for grasping more detailed behavior via the derivative, which can be thought of as the 'instantaneous' rate of change.

Differentiability refers to the ability of a function to have a derivative at every point within a certain interval. When a function is differentiable on an interval, it means that its derivative exists at every point in that interval, indicating that the function has a tangent line that touches the curve exactly at any given point within the interval.

A function being differentiable also implies that it is smooth and has no sharp corners or discontinuities. This concept is critical when dealing with the Mean Value Theorem, which necessitates a function to be differentiable in its open interval domain. When we look at a function such as \( f(x) = \ln (x-1) \), establishing its differentiability on the interval is essential to finding the specific point at which the function's instantaneous rate of change matches its average rate of change over the interval.

Continuity is a characteristic of a function that implies it can be drawn without lifting the pencil from the paper, meaning no gaps, jumps, or sudden changes in direction. A function is considered continuous at a point if the limit of the function as it approaches the point from both directions equals the function’s value at that point.

For a function to fulfill the requirements of the Mean Value Theorem, it must be continuous over the closed interval [a, b]. This is because the theorem relies on the fact that the function behaves predictably over the entire interval. The continuity of functions like \( f(x) = \ln (x-1) \) is one of the prerequisites that need to be checked before we can apply the Mean Value Theorem to find the values of c in the given interval where the instantaneous rate of change equals the average rate of change.