$f(a)$ is defined; $\lim_{x \to a} f(x)$ exists; $\lim_{x \to a} f(x) = f(a)$

$f(a)$ is defined; $f'(a)$ exists; $f$ is bounded near $a$

$\lim_{x \to a} f(x)$ exists; $f$ is bounded near $a$

$f(a)$ is defined; $f'(a)$ exists; $\lim_{x \to a} f(x) = f(a)$