1.11 Defining Continuity at a Point

Learning objectives

—LIM-2.A—Justify conclusions about continuity at a point using the definition—Topic 1.11• LIM-2.A Justify conclusions about continuity at a point using the definition.• LIM-2.A.2 A function f is continuous at x = c provided that f(c) exists, lim x→c f(x) exists, and lim x→c f(x) = f(c).• Enduring understanding LIM-2: Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

A function $f$ is continuous at $x=c$ only when its actual value at $c$ agrees with the single value approached by nearby outputs from both sides.

f(c)\text{ exists},\qquad \lim_{x\to c}f(x)\text{ exists},\qquad \lim_{x\to c}f(x)=f(c).

Check	What failure means
$f(c)$ exists	The function has no defined value at the point
$\lim_{x\to c}f(x)$ exists	The two sides do not approach one common finite value
$\lim_{x\to c}f(x)=f(c)$	Nearby behavior and the point value do not match

Let $f(x)=x+1$ for $x\ne2$ and $f(2)=3$ . First, $f(2)=3$ exists. Second, $\lim_{x\to2}f(x)=\lim_{x\to2}(x+1)=3$ . Third, the limit equals $f(2)$ . All three conditions hold, so $f$ is continuous at $x=2$ .

No single condition is enough. A defined point may sit away from the nearby trend, and a finite limit may exist where the function is undefined. State and verify all three conditions rather than saying only that the graph “has no break.”

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—1.12 Confirming Continuity over an IntervalRead