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I am misunderstanding the diff/continuity definitions in multivariable.

If you take a look here How to show differentiability implies continuity for functions between Euclidean spaces

the original post states that $$\lim_{h \to 0} \frac{\|f(a+h) - f(a) - \lambda(h)\|}{\|h\|} = 0$$

in my definitions I have that $$\lim_{\|h\| \to 0} \frac{\|f(a+h) - f(a) - \lambda(h)\|}{\|h\|} = 0$$

which one would be correct? for instance to say a function is continuous at a point $a$ would we say that $$\lim_{h\to 0} f(a+h) = f(a)$$ or $$\lim_{\|h\| \to 0} f(a+h) = f(a)$$?

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The two are equivalent, I believe. The only way for the norm of $h$ to go to zero is for $h$ to go to zero.

Matt Dickau
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    Just to be really pedantic: As @DanielFischer stated in a comment, it's not that one convergence implies the other one by some theorem, it's that this is the definition of convergence in a metric space. See Wikipedia: "A sequence $(x_n)$ in a metric space M is said to converge to the limit $x \in M$ iff for every $\varepsilon>0$, there exists a natural number $N$ such that $d(x_n,x) < \varepsilon$ for all $n > N$." (Where $d(x_n,x)=\lVert x_n-x\rVert$ in spaces with a metric induced by a norm.) – Piwi Oct 09 '15 at 21:32