Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (l)/Solution 1

None of these have to be true! The fact that the limit exists does not imply anything about the value at that point. In fact, $f(x)$ need not even be defined at $a$. However, if $f(x)$ is not defined at $a$ (C), then writing $f(a)$ does not make sense because this value does not exist (D). $f(x)$ can not be continuous at a point where it's not defined (B), and a function can only be differentiable at those points where it is defined and continuous (A). Therefore, the following is a counter example

Hence, the correct answer is (E).