Making a Function Continuous In Exercises 59-64, find the...

Making a Function Continuous In Exercises $59-64,$ find the constant $a,$ or the constants $a$ and $b$ such that the function is continuous on the entire real number line.
$$f(x)=\left\{\begin{array}{ll}{x^{3},} & {x \leq 2} \\ {a x^{2},} & {x>2}\end{array}\right.$$

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Key Concepts

Continuity

A function is continuous if there are no gaps, jumps, or abrupt changes in its graph. This means that at every point in its domain, the limit of the function as it approaches that point is equal to the function's value at that point.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. To ensure continuity in such functions, especially at the boundaries where the definitions change, the function values and their limits must agree.

Limits

The concept of a limit describes the behavior of a function as its input approaches a particular point. In the context of ensuring continuity, the left-hand limit and right-hand limit at the boundary point must be equal to each other and to the function's value at that point.

Matching Boundary Values

For a piecewise function to be continuous at the point where its definition changes, the function must have equal values when approached from all corresponding intervals. This often leads one to equate the outputs from different expressions at the boundary and solve for any unknown constants.

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