$M \rightarrow M^T M$ is a continuous mapping. Consider the following mapping over $n \times n$ matrices $$f:M \rightarrow M^T M$$. 
My teacher took for granted that it is continous. Intuituively, I'd say that $f$ is a polynomial with the entries of $M$ as variables. That doesn't sound rigorous enough. 
Note that this is unfortunately not a linear mapping. 
I failed finding a proof with $\epsilon$ (I think an operator norm fit best this problem). 
Thanks for any hint.
 A: your first intuition is good: The map f is really polynomial with the entries of $M$ as variables!
I mean, you speak of continuity, right? With respect to which topology? I assume that you are using the identification of the space of $n\times n$ matrices with $\mathbb R^{n^2}$ (if using $\mathbb R$ as coefficient). Therefore the variables are the entries of $M$ and in such coordinates $f$ is polynomial.
A: You talking matrices with entries in $ℝ$ here, right? Then the usual topology of the space $E = \mathrm{Mat}_{n,n} (ℝ)$ of all square matrices is the same as the topology induced by an abovious linear isomorphism $α$ for $E \cong ℝ^{n×n} \cong ℝ^{n^2}$. In other words, such an $α$ defines a homeomorphism.
Therefore, to show that a map $f \colon E → E$ is continuous, it suffices to show that $E \overset{f}{→} E \overset{α}{→} ℝ^{n^2} \overset{π_i}{→} ℝ$ is continuous for all $i = 1, …, n^2$, where $π_i$ denotes the projection to the $i$-th coordinate. Then it would follow that $α ∘ f$ is continuous and so would be $f = α^{-1} ∘ α ∘ f$. So your idea works. In your case $π_i ∘ α ∘ f$ is looking at the $i$-th entry (for some enumeration $i = 1, …, n^2$ of the entries).
A: Both functions transposition
$$t :M \mapsto M^T$$
and matrix multiplication
$$m: (A,B) \mapsto AB$$
are continuous, indeed $t$ is just a permutation of variables, and the only operations involved to compute $m$ are scalar multiplication and sums, that are continuous. 
Thus 
$$f = m \circ (t,id)$$ 
is continuous, where of course $id(A) = A$ :)  
A: hint: you can consider this mapping between two manifolds of dimension $n^2$, there is more general definition for "continuous".
