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A differentiable function is one that has a derivative at every point in its domain. To determine if a function is differentiable at a specific point, we check if the derivative exists there. The derivative at a point represents the slope of the tangent line to the function at that point, giving us a precise rate of change. If a function is differentiable at a point, it means that it has a well-defined tangent there. However, this property also implies something more: it guarantees continuity at that same point. This is because, for a derivative to exist, the function must not have any abrupt changes or discontinuities right there. Remember, while all differentiable functions are continuous, not all continuous functions are differentiable. An example might be the absolute value function, which is continuous everywhere but not differentiable at zero due to the sharp turn.

The concept of the derivative is fundamental in calculus. It is a measure of how a function changes as its input changes.More formally, the derivative of a function at a point gives us the instantaneous rate of change or the slope of the tangent line at that point. Mathematically, if we have a function, say, \(f(x)\), the derivative at a point \(c\) can be defined using the limit:\[ f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \]This expression involves taking the difference quotient and determining its limit as \(x\) approaches \(c\). The existence of this limit confirms that the function is differentiable at \(c\), and hence, continuous.

Continuity of a function at a point implies it does not "break" or have any gaps at that point.For a function \(f(x)\) to be continuous at a point \(c\), it must satisfy three conditions:- \(f(c)\) must be defined- \(\lim_{x \to c} f(x)\) must exist- \(\lim_{x \to c} f(x) = f(c)\)This means the function's limit as it approaches \(c\) is simply the value of the function at \(c\). This seamless transition ensures there are no jumps or breaks in the graph of the function at \(c\).In the exercise provided, the differentiability of a function at \(c\) directly leads to it being continuous there, showing the deep connection between these two concepts.

The difference quotient is a critical concept that acts as a stepping stone towards understanding derivatives. Essentially, it measures the average rate of change of the function over a small interval.For a function \(f(x)\), the difference quotient between two points, close to \(c\), is expressed as:\[ \frac{f(x) - f(c)}{x - c} \]This expression represents how much \(f(x)\) changes as \(x\) changes by some small amount \((x-c)\).As we allow this interval to shrink, meaning \(x\) gets infinitely close to \(c\), the difference quotient approaches the derivative. In the context of differentiability, if this quotient's limit exists as \(x\) nears \(c\), it suggests that the function is smooth, and crucially, continuous at that point. Understanding and calculating the difference quotient lays the groundwork for grasping the concept of derivatives.