Describe the interval(s) on which the function is...

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. $$ f(x)=\left\{\begin{array}{ll}{3+x,} & {x \leq 2} \\ {x^{2}+1,} & {x>2}\end{array}\right. $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied.
$$
f(x)=\left\{\begin{array}{ll}{3+x,} & {x \leq 2} \\ {x^{2}+1,} & {x>2}\end{array}\right.
$$

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Key Concepts

Continuity at a Point

A function is continuous at a point if the function is defined at that point, the limit as the variable approaches the point exists, and the value of the function at the point equals this limit. This three-part condition ensures that there is no disruption in the graph of the function at that specific point.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. Each piece may be a continuous expression on its own, but the overall function is continuous only if the pieces connect smoothly at the transition points, meaning the limits from the left and right at these points agree with the function’s value.

One-sided Limits

One-sided limits examine the behavior of a function as the variable approaches a point from one specific side—either from the left or from the right. This concept is critical when evaluating the continuity of piecewise functions at the points where the formula changes, ensuring that both the left-hand and right-hand limits match the function's value at that point.

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