# Graph and Contour Plots of Functions of Two Variables

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Visualizing the graph of a function of two variables is a useful device to help understand how a function behaves. For a function of two variables , the graph is a surface in 3D space.

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Contributed by: Ana Moura Santos and João Pedro Pargana (March 2011)

Open content licensed under CC BY-NC-SA

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The graph of a function of two variables helps to understand the continuity of the function defined on a domain of . Polynomials of two variables are good examples of everywhere-continuous functions. Here we give an example of the polynomial defined on . In this case the graph consists of a nondegenerate or degenerate quadratic surface. For , , or it is a second-degree polynomial. For , , and it is a plane in 3D. Looking at the corresponding contour plots (a 2D projection of the 3D graphs), gives a better feeling of the behavior of the function.

We know that a composition of two continuous functions is itself a continuous function. Thus a composition of a trigonometric function with a polynomial, in our example , defined on the same domain , is continuous on . In general, the composition of a logarithmic function with a polynomial is not well defined when the argument of the logarithm is negative. Here, is not always well defined on ; try to find out for which values of the constants this happens! Once the function is restricted to a new domain, we have continuity.

In the case of a rational function like , the point is a critical one. One can include this point in the domain of the function and study the limit of the function at . Then, if the limit at this point exists, we have a removable discontinuity. If not, the discontinuity at is not removable. Experiment with different constants and observe that the graph near the point is highly dependent on the coefficients of the numerator. For some choice of the constants, for example and , we have a limit at the origin (but a small "peak" there); on the other hand, for , , and , the graph near the origin is not smooth at all and in the contour plot, different contour lines join together there.

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