Uniform continuity of the min of two continuous functions

Question

Let $(X, M, *)$ be a fuzzy metric space with continuous t-norm $*$. And $f\colon(X, M, *)\rightarrow \mathbb{R}$ is a function defined by $f(x)= \max\{M(x, A, t), \alpha\}$, where $\alpha>0$ and $A$ is any closed set. I have prove that $f$ is continuous. But i am not able to prove that $f$ is uniformly continuous. I have tried to prove it by taking simply $|f(x)-f(y)|,$ but the problem which arise is as how to deal with continuous t-norm and the triangular inequality of the fuzzy metric.