If $f'''(x)$ exists on an interval $[a,b]$, does that mean $f(x)$ is continuous on $[a,b]$?

Question

Does this follow trivially from the fact that differentiability implies continuity, and if $f'''(x)$ exists, then $f(x)$ is differentiable and therefore continuous?

Answer 1

$f'''(x)$ exists $\rightarrow$ $f''(x)$ is differentiable $\rightarrow$ $f''(x)$ is continuous $\rightarrow$ $f'(x)$ is continuously differentiable $\rightarrow$ $f'(x)$ is continuous $\rightarrow$ $f(x)$ is continuously differentiable $\rightarrow$ $f(x)$ is continuous

Answer 2

If $f'''$ exists over $[a,b]$, then $f''$ is differentiable over $[a,b]$, then $f'$ is differentiable over $[a,b]$, then $f$ is differentiable over $[a,b]$, then $f$ is continuous over $[a,b]$.

Answer 3

If $f'''(x)$ exist then $f''(x)$ is differentiable, therefore, continuous. Because $f''(x)$ exists then $f'(x)$ is differentiable, thus, $f'(x)$ is continuous. Because $f'(x)$ exists, then $f(x)$ is differentiable, therefore, $f(x)$ is continuous. This result says that not only $f$ is continuous. The derivatives of $f$ (second and first derivative) are continuous too.