Suppose $f'(x)$ exists for all $x$ in $(a,b)$.

Mark all true items with a check. There may be more than one correct answer.

A. $f(x)$ is continuous on $(a,b)$.

B. $f(x)$ is continuous at $x=a$.

C. $f(x)$ is defined for all $x$ in $(a,b)$.

D. $f'(x)$ is differentiable on $(a,b)$.

In the answer box below, explain your reasoning for each of the three choices you made above. Use complete sentences and correct grammar, spelling, and punctuation. Be specific and detailed. Write as if you were explaining the answer to someone else in class.