Functions That Agree at All But One Point

Let $c$ be a real number and let $f(x)=g(x)$ for all $x\ne c$ in an open interval containing $c$ . If $\lim_{x\to c}g(x)$ exists, then $\lim_{x\to c}f(x)$ also exists and is equal to $\lim_{x\to c}g(x)$ .