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Continuity is a fundamental concept in calculus that describes a function that has no breaks, holes, or jumps in its graph. A function is continuous at a point if the function's value at that point is equal to the limit of the function as it approaches that point from both directions.
For example, consider the function \( f(x) = |x| \). This function is continuous at every point in its domain, the set of all real numbers. This means that for any point on the graph of \( f(x) \), if you zoom in closely, it appears as an unbroken curve, without any gaps or abrupt changes.

• The function is defined at every point.
• The left-hand limit and the right-hand limit at each point exist and are equal.
• The value of the function at a point is the same as the limits as you approach that point.

Understanding continuity is crucial as it ensures that a function behaves in a predictable way over its entire domain, making analysis simpler and more intuitive.

Differentiability is a property of a function that indicates if it has a defined derivative at a particular point. A function is differentiable at a point if it can be smoothly approximated by a tangent line at that point, which means there are no corners or cusps.
Considering the function \( f(x) = |x| \), we find it is differentiable everywhere except at \( x = 0 \). At \( x = 0 \), the graph has a sharp corner, creating a point where the slope is not well-defined. Differentiability requires the following:

• The function must be continuous at the point of interest.
• The slope (or rate of change) must approach the same value from the left and the right.

At \( x = 0 \), the slopes of the lines described by \( y = x \) and \( y = -x \) don't match as you approach zero from either side of the "V" shape, hence the function is not differentiable there. Differentiability implies continuity, but the reverse is not always true.

A piecewise function is defined by different expressions depending on the input value. It allows for the construction of functions that behave differently over various parts of their domain. This is especially useful for designing graphs with specific characteristics, like the one described in the original exercise.
The function \( f(x) = |x| \) is an example that can be expressed as a piecewise function. It consists of:

• \( y = x \) for \( x \geq 0 \)
• \( y = -x \) for \( x < 0 \)

At \( x = 0 \), the two "pieces" connect smoothly at the origin, forming a continuous function. However, the sharp transition at this point results in it being not differentiable there.
Piecewise functions are powerful tools in modeling scenarios where the behavior changes based on certain conditions, making them widely applicable in various real-world situations.