Closed graph theorem

The graph of the cubic function ${\displaystyle f(x)=x^{3}-9x}$ on the interval ${\displaystyle [-4,4]}$ is closed because the function is continuous. The graph of the Heaviside function on ${\displaystyle [-2,2]}$ is not closed, because the function is not continuous.

In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.

A T. Tao’s blog post^[1] lists several closed graph theorems throughout mathematics.