Reference Request: Modulus of Continuity of Matrix Multiplication

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Let $M:GL_n\times \mathbb{R}^n\rightarrow \mathbb{R}^n$; equip $\mathbb{R}^n$ with its usual Euclidean topology and $GL_n$ with its relative Euclidean topology (from the Frobenius norm). Is the modulus of continuity of this function known?