The closed graph theorem for nonlinear maps (Q1289036)

The closed graph theorem alluded to in the title can be stated as follows: Let \(X\) be a real vector space with a translation-invariant complete metric \(d\) satisfying \(d(\frac 1n x,0)\to 0\) as \(n\to\infty\) for all \(x\in X\) and \(d(\frac 1m x_n,0)\to 0\) as \(n\to\infty\) whenever \(m\) is a positive integer and \(d(x_n,0)\to 0\). Let \((Y,\rho)\) be a complete metric space and \(f:X\to Y\) a function with closed graph satisfying \(\lim_{n\to\infty}f(x+\frac 1n u)=f(x)\) uniformly in \(x\in X\) whenever \(u\in X\). Then \(f\) is uniformly continuous.