# Does the boundedness of partial derivatives imply continuity?

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

Does this imply that the given function is continuous?

### Remark

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 $xy/(x^2+y^2)$.

## 1 Answer

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 ).\$