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Continuity is a concept in calculus that describes a function that has no breaks, jumps, or holes in its graph. A function is continuous over an interval if you can draw its graph without lifting your pen from the paper. This means the function is well-defined and smooth across that range.
For the given function, \( f(x) = \sqrt{x-1} \), we notice that it involves a square root. A square root function is continuous for all values where the expression inside the square root is non-negative. Here, \( x-1 \geq 0 \) makes \( x \geq 1 \). Since the interval \([2,5]\) lies completely within this domain, we can comfortably say our function is continuous on this interval. The absence of any gaps assures us that we can apply the Mean Value Theorem if other conditions are met.

Differentiability is a condition that ensures a function has a derivative at every point in the interior of an interval. For a function to be differentiable, its derivative must exist and be finite at each point within the interval, leading to a 'smooth' curve.
The function \( f(x) = \sqrt{x-1} \) is differentiable for \( x > 1 \). This is because the function's derivative, which tells us the rate of change of \( f \) with respect to \( x \), is defined and can be calculated in this domain. The interval \((2,5)\) fits perfectly within \(x > 1\), confirming that \( f \) is differentiable on this open interval. Differentiability is crucial for applying the Mean Value Theorem because it assures us that the function's graph can be approximated by lines (tangents) at every point in the interior.

The derivative of a function is a measure of its rate of change. It shows how the function's output changes with slight changes in its input, and is represented as \( f'(x) \).
For the function \( f(x) = \sqrt{x-1} \), we find its derivative using the power rule, transformed slightly due to the square root. The derivative is calculated as \( f'(x) = \frac{1}{2\sqrt{x-1}} \). This tells us that as \( x \) increases, the slope of the tangent to the function's curve changes, highlighting how rapidly the function's value grows. This derivative is essential for computing the specific value \( c \) that satisfies the Mean Value Theorem, linking the function's average rate of change on an interval to its instantaneous rate of change at a point within that interval.

An interval in mathematics refers to a range of values, usually defined by two endpoints. It can be open (excluding endpoints) or closed (including endpoints).
In our example, we have the interval \([2,5]\). This denotes all numbers \( x \) satisfying \( 2 \leq x \leq 5 \). The interval provides the domain over which we analyze the function \( f(x) = \sqrt{x-1} \). For the Mean Value Theorem, both continuity on the closed interval and differentiability on the open interval are key. This means the conditions are checked for \([2,5]\) regarding continuity and for \((2,5)\) regarding differentiability. Understanding intervals helps us determine where a certain mathematical rule like the Mean Value Theorem can be applied, ensuring that the function behaves predictably across specified boundaries.