Suppose that the partial derivatives of a given function $f$ are bounded.

Does this imply that the given function is continuous?


For functions of one variable this is true, because every differentiable function is continuous. But in several variables there are discontinuous functions which have partial derivatives everywhere, like $xy/(x^2+y^2)$.


Let $f:V=(x_0 -h ,x_0 +h)\times (y_0-h ,y_0 +h)\to \mathbb{R},$$h>0$ be a function of two variables. Suppose that there exists $M>0$ such that $$\sup_{(u,w)\in V}\max\left\{\left|\frac{\partial f}{\partial x} (u,v)\right| ,\left|\frac{\partial f}{\partial y} (u,v)\right|\right\} \leq M$$ then $f$ is continuous at $(x_0 ,y_0 ).$

To see this , take any $(\sigma , \xi )\in V$ then using Lagrange mean value theorem we have: $$|f(x_0 , y_0 )- f(\sigma , \xi ) |\leq |f(x_0 , y_0 ) -f(x_0 , \xi )|+|f(x_0 , \xi ) -f(\sigma , \xi )| \leq M(|y_0 -\xi|+|x_0 -\sigma |)$$ which implies continuity at $(x_0 ,y_0 ).$


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