The graph of a function of two variables helps to understand the continuity of the function defined on a domain
. Polynomials of two variables are good examples of everywhere-continuous functions. Here we give an example of the polynomial
. In this case the graph consists of a nondegenerate or degenerate quadratic surface. For
it is a second-degree polynomial. For
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
, we have a limit at the origin (but a small "peak" there); on the other hand, for
, the graph near the origin is not smooth at all and in the contour plot, different contour lines join together there.