For the standard definition of Lipschitz continuity we have $$||f(x) - f(y)|| \leq L||x-y||$$
But what do we do if we define $$f(a,b) = \gamma_1(a) + \gamma_2(b) : a \in R^{n \times m}, b \in R^{n \times k} $$ $\gamma_1, \gamma_2$ both have Lipschitz constants $\gamma_{1_L},\gamma_{2_L}$ respectively.
How can we define the Lipschitz constant for $f$?
First, the answer depends on the choice of norm/metric on $\mathbb{R}^{n\times m} \times \mathbb{R}^{n\times k}$. Now let's estimate:
$$\|f(a,b)-f(c,d)\| = \|\gamma_1(a)+\gamma_2(b) - \gamma_1(c)-\gamma_2(d)\|,$$
so rearranging and using the triangle inequality we obtain
$$\|f(a,b)-f(c,d)\| \leq \|\gamma_1(a)-\gamma_1(c)\| + \|\gamma_2(b)-\gamma_2(d)\| \\\leq \text{Lip}(\gamma_1) \|a-c\| + \text{Lip}(\gamma_2)\|b-d\|.$$
Hence
$$\|f(a,b)-f(c,d)\| \leq \max\{\text{Lip}(\gamma_1),\text{Lip}(\gamma_2)\}\left(\|a-c\| + \|b-d\|\right).$$
If you use the norm $\|(a,b)\|:=\max\{\|a\|,\|b\|\}$ on $\mathbb{R}^{n\times m} \times \mathbb{R}^{n\times k}$, then $L:= \max\{\text{Lip}(\gamma_1),\text{Lip}(\gamma_2)\}$ is a Lipschitz constant for $f$. If you use a different norm, then you need to do a little more work.