Does $f\in C^1$ Implies $f\in C$

Question

Does the fact that the partial derivates are continuous and therefore the function is differentiable we can conclude that the function is continuous (differentiability $\rightarrow$ continuity )?

Answer 1

$\|f(x+h)-f(x)\|\leq\|h\|^{-1}\|f(x+h)-f(x)-Df(x)h\|\|h\|+\|Df(x)\|\|h\|\rightarrow 0$, so $f$ is continuous at $x$.