Determine whether each statement makes sense or does not...

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
If $\lim _{x \rightarrow a^{-}} f(x) \neq f(a)$ and $\lim _{x \rightarrow a^{-}} f(x) \neq \lim _{x \rightarrow a^{+}} f(x)$, I can redefine $f(a)$ to make $f$ continuous at $a$.

Key Concepts

Continuity at a Point

Continuity at a point requires that the function’s limit (as x approaches the point from both sides) exists and equals the function’s value at that point. In an ideal continuous function at a point, both the left-hand and right-hand limits must exist, be equal, and match the function's value at that point. If these conditions are not met, the function is said to be discontinuous at that point.

Removable Discontinuities

A removable discontinuity occurs when a function’s limit exists at a point, but the function either is not defined at that point or is defined to be a different value than its limit. In such cases, redefining the function at the point can 'remove' the discontinuity, resulting in a continuous function at that point. However, if the left-hand and right-hand limits do not agree, the discontinuity is not removable, because no single redefinition can reconcile the differing limits.

One-Sided Limits

One-sided limits involve evaluating the behavior of a function as the input approaches a particular point from one side only—either from the left (x approaches a from values less than a) or from the right (x approaches a from values greater than a). Understanding one-sided limits is essential because they help determine whether the function approaches the same value from both directions, which is a prerequisite for overall continuity at that point.

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