Answer
See explanations.
Work Step by Step
Step 1. Given $\lim_{h\to0}f(c+h)=f(c)$, let $x=c+h$. Then, when $h\to0$, $x\to c$, the limit becomes $\lim_{x\to c}f(x)=f(c)$
Step 2. The above result indicates that (i) $f(c)$ exists, (ii) $\lim_{x\to c}f(x)$ exists, and (iii) $\lim_{x\to c}f(x)=f(c)$
Step 3. The above three conditions complete the if-and-only-if Continuity Test of a function and we end our proof.
