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A continuous function is one that you can draw without lifting your pen off the paper. This means there are no sudden jumps, breaks, or holes within the interval. More technically, a function is continuous on an interval \[ [a, b] \] if for every point \((x_0)\) within the interval, the limit as \(x\) approaches \(x_0\) is exactly equal to \((f(x_0))\).
This property is crucial because it creates a predictable behavior of the function throughout that interval. A continuous function does not suddenly spike or drop between two points. This property allows us to apply the Mean Value Theorem and other calculus tools to study the function further.
For example, with the problem at hand, we're assessing a continuous function on the interval \[ [a, b] \]. This continuity ensures that the Mean Value Theorem can be applied because the theorem requires the function to be continuous on a closed interval and differentiable on the open interval.

A differentiable function is where the derivative exists for every point within a chosen domain. Essentially, it means that at any point on the curve, you can compute a tangent line, providing a slope for the curve at that point.
For the function \(f\), being differentiable on the interval \( (a, b)\) implies smoothness. There are no sharp corners or cusps within \( (a, b)\).
It's important to understand this property because it guarantees the existence of `f'(x)` within that interval, ensuring we can analyze the function's behavior based on its slopes. In this particular exercise, since the derivative \( f'(x) < 0\), we can infer that the function is decreasing right through the interval. The negative derivative suggests that the slope of the function is downward, bolstering our understanding of the function's monotonic behavior in this interval.

The Mean Value Theorem is a fundamental principle in calculus that connects derivatives with differences in values of a function. It states that if a function \(f\) is continuous on \[ [a, b] \] and differentiable on \( (a, b)\), there is at least one point \(c\) within that interval where the derivative at \(c\) equals the function's average rate of change over \[ [a, b] \].
Mathematically, this is represented as: \[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
In our context, applying the Mean Value Theorem helps confirm that the function is decreasing. If \( f'(x) < 0\) for all \( x \) in \( (a, b)\), then the slope of the line connecting any two points within \( f \) is negative, establishing that \( f(x_2) < f(x_1)\) for \( x_1 < x_2\). This theorem crystallizes the understanding that if the derivative is consistently negative, the entire function descends, aligning with the definition of a decreasing function.