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Rope tension definition

What is the exact definition of the tension of a rope?

Here http://web.mit.edu/8.01t/www/materials/modules/chapter08.pdf the following definition is given

The tension $T(x)$ in a rope at a distance $x$ from one end of the rope is the magnitude of the action -reaction pair of forces acting at the point $x$

But I think that this is valid only in the case that the tension is the same, throughout all the rope. Is that definition still valid in that case?