Chapter 1.5, Problem 43ES

(a) Use Theorem $\mathrm{1.5.5}$ to prove that if $f$ is continuous at $x=c$ , that ${\lim }_{h\to 0}f\left(c+h\right)=f\left(c\right)$ .

(b) Prove that if ${\lim }_{h\to 0}f\left(c+h\right)=f\left(c\right)$ , then $f$ is continuous at $x=c$ .

(c) Conclude from part (a) and (b) that $f$ is continuous at $x=c$ if and only if ${\lim }_{h\to 0}f\left(c+h\right)=f\left(c\right)$ .