Continuity

A function $f$ is continuous at c if the following three conditions are met:

$$f(c)\;$$

$$\lim_{x\to c} f(x)\;$$

$$\lim_{x\to c}f(x)=f(c).$$

A function $f$ is continuous on an open interval $(a,b)$ if it is continuous at each point in the interval. A function $f$ is continuous on a closed interval $[a,b]$ if it is continuous on the open interval $(a,b)$ and

$$\lim_{x\to a^+}f(x)=f(a) \quad\lim_{x \to b^-}=f(b).$$