If $f'''(x)$ exists on an interval $[a,b]$, does that mean $f(x)$ is continuous on $[a,b]$? Does this follow trivially from the fact that differentiability implies continuity, and if $f'''(x)$ exists, then $f(x)$ is differentiable and therefore continuous?

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 $[a,b]$. 
