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Piecewise functions are mathematical expressions defined by multiple sub-functions, each applicable to a specific interval of the main function's domain. These functions display different behaviors depending on the range of input values they receive.
For example, in the given function:

• If \( x < -1 \), the expression is \( \frac{x^3 - x}{x^2 - 1} \).
• If \(-1 \leq x \leq 2\), the expression is \( 2x + 1 \).
• If \( x > 2 \), the expression is \( \frac{x^2 - 4}{x - 2} \).

Switching to different formulas regarding the intervals results in the piecewise nature. The critical task is how to manage the transitions between these sub-functions smoothly.
This can be particularly delicate at points where the piece transitions, like at boundaries \( x = -1 \), \( x = 1 \), or \( x = 2 \). Ensuring continuity involves verifying specific conditions, which can turn these math pieces into smoothly traversed parts of a function.

Limit evaluation in calculus is crucial when determining the behavior of functions as they approach certain points or infinity. It examines the values a function tends towards as the variable gets closer to a particular point.
The steps to evaluate a limit often involve substitution or algebraic manipulation to find an expression that works for the entire domain except the point in question. This can require different techniques for piecewise functions.
For example, to find \( \lim_{{x \to -1^-}} \) of \( \frac{x^3 - x}{x^2 - 1} \), one might simplify or rearrange the expression to handle direct computation or decompose it to remove undefined operations.
Limit analysis helps determine whether a function is continuous at a point by checking if the left-hand and right-hand limits equal and match the function's value at that point (if defined). Thus, mastering limit evaluation is key to grasping function behavior near those crucial transition points.

Discontinuity points are specific input values where a function isn't continuous. At these points, the function fails the basic continuity checks, which include:

• The function is defined at the point of interest.
• The left-hand limit equals the right-hand limit at that point.
• Both limits (from either side) are equal to the value of the function at the point.

When even one of these conditions isn't met, the function is discontinuous at that location. Discontinuity can be of various types:

• Removable: If the function can be redefined at that point to "fix" the discontinuity.
• Jump: If the left and right-hand limits exist but aren't equal.
• Infinite: If the function grows indefinitely at that point.

In our example, \( x = -1 \) and \( x = 2 \) are discontinuity points. At \( x = -1 \), the function isn't defined from the left side. At \( x = 2 \), a jump discontinuity exists because the limits from either side don't equal each other, nor do they match \( f(2) \). Understanding these classifications helps in figuring out how a function behaves and what kind of corrections, if any, can bring about continuity.