Dirk Bollaerts &
Céleste Paalvast

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Exercises on the continuity of functions of two variables

1. Overview of continuity of functions of two variables

  • Section 1.
    We discuss two standard exercises on continuity and discontinuity of real functions in real variables.
  • Section 2.
    We list the definitions of the functions discussed in all the exercises that can be found in the text document that the reader can download.
  • Section 3.
    The reader can download the text document in the download section.
  • Section 4.
    The reader can find information about web based material in the information section about links and search material .

2. Continuity of functions in two real variables: exercise notes

We discussed in the past with students a few standard examples of exercises about continuity and discontinuity of real functions of two variables and have collected the exercises in a text document. This text can be downloaded. Some examples of exercises are standard examples and can be found in many texts.

We think that the text can be useful for everyone studying the theory of functions in two variables for the first time. The text is certainly not aimed only at students with a mathematics major. We think that the general public studying a calculus text can make use of it.

2.1 Two standard exercises

Exercise 1. (Exercise on discontinuity).

Is the function f(x,y)={xyx2+y2if (x,y)(0,0);0if (x,y)=(0,0) continuous in (0,0)?

Solution.

Let us calculate the function restricted to a line y=λx. We have in that case:

f(x,λx)={λx2x2+λ2x2=λ1+λ2if x0;0if x=0.

We calculate the limits. limx0f(x,λx)=λ1+λ2. This gives us an infinite amount of different limit values for the function restricted to the different lines. If the function is continuous, then all these values should give us the function value in (0,0), that is f(0,0)=0. So this function is not continuous.

Surface.  Clear view of the Discontinuity.
We see here a three dimensional figure of the graph of the function. We see that the function above the point (0,0) shows a vertical line. The reason for this is that the limit points of the level curves are unavoidably visible above (0,0). The graph does not show a function any more, because it looks like (0,0) is mapped onto many points. But it seems to be unavoidable. Otherwise we could show nothing at all.
Surface.  Clear view of the function restricted to lines.
We have restricted the function here to y=12x and y=310x and y=910x. We see in this figure clearly that the restrictions of the function to these lines are functions that have different limits in 0. This cannot be the case if the function is continuous.
Level lines of the surface.
We see here a figure of the contour plot of the function. Remark that level curves of very different levels approach (0,0) infinitesimally. This is always very suspicious and it is almost certain or at least a very strong indication that the function is not continuous in (0,0). The level curves plot shows a very peculiar picture. It consists of straight lines through the origin. This gives us a picture of a spiral staircase in which the lines represent the steps and the vertical line above (0,0) is the supporting vertical beam of the spiral staircase. This cannot be a picture of a continuous point.
Exercise 2. (Exercise on continuity).

Is the function f(x,y)={(2x+y)3x2+y2if (x,y)(0,0);0if (x,y)=(0,0) continuous in (0,0)?

Solution.
We take an arbitrary ϵ>0. The problem is now to find a δ>0 such that if ||(x,y)(0,0)||<δ holds then |f(x,y)f(0,0)|<ϵ is valid.

By looking at our function, we have in our case the following statements. Try to find a δ such that if ||(x,y)(0,0)||=x2+y2<δ, we have that

|(2x+y)3x2+y20|<ϵ.

We are looking for a function (x,y) that is larger then or equal to the left hand side of this last inequality. This function (x,y) has to have the property that it can be made smaller then ϵ by carefully manipulating the value δ. This is sufficient for our continuity proof.

|(2x+y)3x2+y20|(|2x|+|y|)3x2+y2(2|x|+|y|)3x2+y2(2x2+y2+x2+y2)3x2+y227x2+y23x2+y2227x2+y2.

It is sufficient to prove that 27x2+y2<ϵ by manipulating the magnitude of δ in the inequality x2+y2<δ. We can choose for example x2+y2<δ with δ=ϵ27. We have found a δ and the function is indeed continuous.

Surface.  Clear view of the 3D view of the continuity.
We see here a three dimensional figure of the graph of the function.
Surface.  Clear view of the the level curves plot of the continuity.
We see here a figure of the contour plot of the function. This looks like a classic picture of a continuous point. We see that the level lines that come close to (0,0) tend to have a level approximating 0.

2.2 List of exercises

Exercise 1.

f(x,y)={xyx2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 2.

f(x,y)={(2x+y)3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 3.

f(x,y)={1x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 4.

f(x,y)={1xyif xy;0if x=y.

Exercise 5.

f(x,y)={x2yx2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 6.

f(x,y)={xy2x2+y4if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 7.

f(x,y)={xy3x2+y4if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 8.

f(x,y)={xy3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 9.

f(x,y)={xycos(y)x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 10.

f(x,y)={ex2+y21x+yif xy;0if x=y.

Exercise 11.

f(x,y)={5x2yy3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 12.

f(x,y)={x3y2x4+y4if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 13.

f(x,y)={sin(xy)x+yif (x,y)(0,0);0if (x,y)=(0,0).

Exercise 14.

f(x,y)={sin(xy)x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 15.

f(x,y)={5x2yy3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 16.

f(x,y)={x2y2x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 17.

f(x,y)={x3+y3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 18.

f(x,y)={x3yx4+y4if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 19.

f(x,y)={x3y3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 20.

f(x,y)={2xyx2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 21.

f(x,y)={(x1)2ln(x)(x1)2+y2if (x,y)(1,0);0if (x,y)=(1,0).

Exercise 22.

f(x,y)={sin((xy)2)|x|+|y|if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 23.

f(x,y)={1x2yif yx2;0if y=x2.

Exercise 24.

f(x,y)={arctan(xy2x+y)if xy;0if x=y.

Exercise 25.

f(x,y)={x2y2x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 26.

f(x,y)={xx2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 27.

f(x,y)={xyx+yif xy;0if x=y.

Exercise 28.

f(x,y)={x2+yyif y0;0if y=0.

Exercise 29.

f(x,y)={x3xy3x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 30.

f(x,y)={x2x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 31.

f(x,y)={x2+y2x3y3x2+y2if (x,y)(0,0);1if (x,y)=(0,0).

Exercise 32.

f(x,y)={x2yx4+y4if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 33.

f(x,y)={x2y2x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 34.

f(x,y)={|x|a|y|b(x2+y2)cif (x,y)(0,0);0if (x,y)=(0,0).

Exercise 35.

f(x,y)={(x+y)sin(1x)sin(1y)if xy0;0if xy=0.

Exercise 36.

f(x,y)={xyx2+y2+ysin(1x)if x0;0if x=0.

Exercise 37.

f(x,y)={xsin(1y)+ysin(1x)if xy0;0if xy=0.

Exercise 38.

f(x,y)={x2y2x3+y3if xy;0if x=y.

Exercise 39.

f(x,y)={xy(x2y2)x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 40.

f(x,y)={x2+sin2(y)2x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 41.

f(x,y)={xyx2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 42.

f(x,y)={sin(xy)x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 43.

f(x,y)={arcsin(xy)(1+xy)if xy1 , ;0elsewhere.

Exercise 44.

f(x,y)=x+yx2+y2+1.

Exercise 45.

f(x,y)={(x1)2ln(x)(x1)2+y2if (x,y)(1,0) and x>0;0elsewhere.

Exercise 46.

f(x,y)={x+yx2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 47.

f(x,y)={x3y3x3+y3if xy;0if x=y.

Exercise 48.

f(x,y)={arctan(|x|+|y|x2+y2)if (x,y)(0,0);π2if (x,y)=(0,0).

Exercise 49.

f(x,y)={ex2y21x2+y2+1if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 50.

f(x,y)={1if xy=0;0if xy0.

Exercise 51.

f(x,y)={xy|x|+|y|if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 52.

f(x,y)={x22yy2+2xif (x,y)(0,0);0if (x,y)=(0,0).

Exercise 53.

f(x,y)={x2y2x4+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 54.

f(x,y)={sin(x2+4y2)x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 55.

f(x,y)={|x|y1if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 56.

f(x,y)={exyyif y0;0if y=0.

Exercise 57.

f(x,y)={sin(xy3)x2if x0;0if x=0.

Exercise 58.

f(x,y)={x3+y5x2+2y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 59.

f(x,y)={x2+sin2(y)x2+2y2if x0;0if x=0.

Exercise 60.

f(x,y)={tan(xyxy+1)xyif xy0;1if xy=0.

Exercise 61.

f(x,y)={(x2+y2)x2y2if (x,y)(0,0);1if (x,y)=(0,0).

Exercise 62.

f(x,y)={(x+1)(yx)(y+1)(x+y)if xy and y1;0if x=y or y=1.

Exercise 63.

f(x,y)={|y|a|x|b|x|a+|y|bif (x,y)(0,0);0if (x,y)=(0,0).

Exercise 64.

f(x,y)={arctan(yx)if x0;0if x=0.

Exercise 65.

f(x,y)={x3y3x6+y6if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 66.

f(x,y)={x3yx6+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 67.

f(x,y)={(x1)y2x2+y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 68.

f(x,y)={x2y6xy3if x0 and y0;0if x=0 or y=0.

Exercise 69.

f(x,y)={arctan(2yx2+y2)if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 70.

f(x,y)={|y|sin(x|y|)if y0;0if y=0.

Exercise 71.

f(x,y)={x2(1cos(yx))if x0;0if x=0.

Exercise 72.

f(x,y)={sin(1xy)if x0 and y0;0if x=0 or y=0.

Exercise 73.

f(x,y)={sin(1x2+y2)if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 74.

f(x,y)={(x2+y2)μsin(1x2+y2)if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 75.

f(x,y)={cos(xy)sin(4x|y|)|xy|if x0 and y0;0if x=0 and y=0.

Exercise 76.

f(x,y)={xnymsin(1x2+y2)if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 77.

f(x,y)={1cos(x2y)x2+2y2if (x,y)(0,0);0if (x,y)=(0,0).

Exercise 78.

f(x,y)={x2yx2y2if xy;0if x=y.

Exercise 79.

f(x,y)={xyx+yif xy;0if x=y.

3. Download the exercises text

The text document “Exercise notes on the continuity of functions of two variables” with all the exercises and solutions can be viewed online and optionally downloaded. This is a file in pdf format. The text document is not small though: the size is about 47 megabytes due to many illustrations. The illustrations have a resolution of the industry standard 270 DPI. This enables the reader to reasonably enlarge it to see more detail. The text will also be ready to be printed on high resolution devices if necessary.

A good place to start is always wikipedia.

Because the subject of continuity in two variables is frequently discussed and continuously changed on the net, we can perhaps advise to use your favourite search machine with a search string of something like continuity functions two variables.

https://www.google.com/search?q=continuity+functions+two+variables

We continue with a few links to physicsforums.com, mathexchange.com and mathoverflow.net. Starting from there, the reader can probably find his own way.

We use our favourite search machine with for example the search string multivariable calculus continuity functions two variables. We can change the search string at leisure.

Here are some websites with explanations about the continuity of functions in two variables.

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