Given a function of two (or more) variables some properties of it can be deduced by assigning some fixed numerical value to all but one variables and treating it as a function of remaining variables only.

For example if is a function of two variables, then for fixed, we get a function of a single variable

And similarly, for fixed, we get the function of a single variable

These functions do not give us complete information about the function For example, both of them may be continuous at and , respectively, but need not be continuous at However, they can give us some useful information about . For example if either of ( or ) is discontinuous at ( or at ), then clearly cannot be continuous at We next look at the differentiability properties of these functions.

Let be an interior point and . To understand how does the surface behave near the point , consider the curve obtained by the intersection the surface with the plane at . This will give us a curve in the plane . One way of understanding near is to analyze this curve. For example we can try to draw tangent to this curve at , i.e., analyze whether the function has a derivative at or not.