Is the gradient of $f(x,t)=xe^{t/x}$ w.r.t. ($x,t$) a Lipschitz continuous function?

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Consider a function $f(x,t)=xe^{t/x}$ where $C \ge x,t\ge0$ with $C$ being a positive constant. Then, is the gradient $\Delta f$ a Lipschitz continuous function over the domain $[0,C]^2$?