Exercises on the continuity of functions of two variables

Exercise 1.

$$\begin{array}{c}f(x,y)=\{\begin{array}{cc}\frac{xy}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}\end{array}$$

Exercise 2.

$$f(x,y)=\{\begin{array}{cc}\frac{(2x+y{)}^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 3.

$$f(x,y)=\{\begin{array}{cc}\frac{1}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 4.

$$f(x,y)=\{\begin{array}{cc}\frac{1}{x-y}& \text{if }x\ne y;\\ 0& \text{if }x=y.\end{array}$$

Exercise 5.

$$f(x,y)=\{\begin{array}{cc}\frac{x^{2}y}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 6.

$$f(x,y)=\{\begin{array}{cc}\frac{x{y}^2}{x^2+y^4}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 7.

$$f(x,y)=\{\begin{array}{cc}\frac{x{y}^3}{x^2+y^4}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 8.

$$f(x,y)=\{\begin{array}{cc}\frac{xy}{3x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 9.

$$f(x,y)=\{\begin{array}{cc}\frac{xy\cos (y)}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 10.

$$f(x,y)=\{\begin{array}{cc}\frac{e^{x^2+y^2}-1}{x+y}& \text{if }x\ne -y;\\ 0& \text{if }x=-y.\end{array}$$

Exercise 11.

$$f(x,y)=\{\begin{array}{cc}\frac{5x^{2}y-y^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 12.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3{y}^2}{x^4+y^4}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 13.

$$f(x,y)=\{\begin{array}{cc}\frac{\sin (xy)}{x+y}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 14.

$$f(x,y)=\{\begin{array}{cc}\frac{\sin (xy)}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 15.

$$f(x,y)=\{\begin{array}{cc}\frac{5x^{2}y-y^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 16.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2{y}^2}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 17.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3+y^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 18.

$$f(x,y)=\{\begin{array}{cc}\frac{x^{3}y}{x^4+y^4}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 19.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3-y^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 20.

$$f(x,y)=\{\begin{array}{cc}\frac{2x-y}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 21.

$$f(x,y)=\{\begin{array}{cc}\frac{(x-1{)}^2\ln (x)}{(x-1{)}^2+y^2}& \text{if }(x,y)\ne (\mathrm{1,0});\\ 0& \text{if }(x,y)=(\mathrm{1,0}).\end{array}$$

Exercise 22.

$$f(x,y)=\{\begin{array}{cc}\frac{\sin ((x-y{)}^2)}{|x|+|y|}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 23.

$$f(x,y)=\{\begin{array}{cc}\frac{1}{x^2-y}& \text{if }y\ne x^2;\\ 0& \text{if }y=x^2.\end{array}$$

Exercise 24.

$$f(x,y)=\{\begin{array}{cc}\arctan \left(\frac{x{y}^2}{x+y}\right)& \text{if }x\ne -y;\\ 0& \text{if }x=-y.\end{array}$$

Exercise 25.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2-y^2}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 26.

$$f(x,y)=\{\begin{array}{cc}\frac{x}{\sqrt{x^2+y^2}}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 27.

$$f(x,y)=\{\begin{array}{cc}\frac{x-y}{x+y}& \text{if }x\ne -y;\\ 0& \text{if }x=-y.\end{array}$$

Exercise 28.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2+y}{y}& \text{if }y\ne 0;\\ 0& \text{if }y=0.\end{array}$$

Exercise 29.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3-x{y}^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 30.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 31.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2+y^2-x^3{y}^3}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 1& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 32.

$$f(x,y)=\{\begin{array}{cc}\frac{x^{2}y}{x^4+y^4}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 33.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2{y}^2}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 34.

$$f(x,y)=\{\begin{array}{cc}\frac{|x{|}^a|y{|}^b}{(x^2+y^2{)}^c}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 35.

$$f(x,y)=\{\begin{array}{cc}(x+y)\sin (1⁄x)\sin (1⁄y)& \text{if }xy\ne 0;\\ 0& \text{if }xy=0.\end{array}$$

Exercise 36.

$$f(x,y)=\{\begin{array}{cc}\frac{xy}{x^2+y^2}+y\sin (1⁄x)& \text{if }x\ne 0;\\ 0& \text{if }x=0.\end{array}$$

Exercise 37.

$$f(x,y)=\{\begin{array}{cc}x\sin (1⁄y)+y\sin (1⁄x)& \text{if }xy\ne 0;\\ 0& \text{if }xy=0.\end{array}$$

Exercise 38.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2{y}^2}{x^3+y^3}& \text{if }x\ne -y;\\ 0& \text{if }x=-y.\end{array}$$

Exercise 39.

$$f(x,y)=\{\begin{array}{cc}\frac{xy(x^2-y^2)}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 40.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2+{\sin }^2(y)}{2x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 41.

$$f(x,y)=\{\begin{array}{cc}\frac{xy}{\sqrt{x^2+y^2}}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 42.

$$f(x,y)=\{\begin{array}{cc}\frac{\sin (xy)}{\sqrt{x^2+y^2}}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 43.

$$f(x,y)=\{\begin{array}{cc}\frac{\arcsin (x⁄y)}{(1+xy)}& \text{if }xy\ne -1\text{ , }\dots ;\\ 0& \text{elsewhere.}\end{array}$$

Exercise 44.

$$f(x,y)=\frac{x+y}{x^2+y^2+1}.$$

Exercise 45.

$$f(x,y)=\{\begin{array}{cc}\frac{(x-1{)}^2\ln (x)}{(x-1{)}^2+y^2}& \text{if }(x,y)\ne (\mathrm{1,0})\text{ and }x>0;\\ 0& \text{elsewhere.}\end{array}$$

Exercise 46.

$$f(x,y)=\{\begin{array}{cc}\frac{x+y}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 47.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3-y^3}{x^3+y^3}& \text{if }x\ne -y;\\ 0& \text{if }x=-y.\end{array}$$

Exercise 48.

$$f(x,y)=\{\begin{array}{cc}\arctan \left(\frac{|x|+|y|}{x^2+y^2}\right)& \text{if }(x,y)\ne (\mathrm{0,0});\\ \pi ⁄2& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 49.

$$f(x,y)=\{\begin{array}{cc}\frac{e^{-x^2-y^2}-1}{x^2+y^2}+1& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 50.

$$f(x,y)=\{\begin{array}{cc}1& \text{if }xy=0;\\ 0& \text{if }xy\ne 0.\end{array}$$

Exercise 51.

$$f(x,y)=\{\begin{array}{cc}\frac{xy}{|x|+|y|}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 52.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2-2y}{y^2+2x}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 53.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2{y}^2}{x^4+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 54.

$$f(x,y)=\{\begin{array}{cc}\frac{\sin (x^2+4y^2)}{x^2+y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 55.

$$f(x,y)=\{\begin{array}{cc}|x{|}^y-1& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 56.

$$f(x,y)=\{\begin{array}{cc}\frac{e^{-\frac{x}{y}}}{y}& \text{if }y\ne 0;\\ 0& \text{if }y=0.\end{array}$$

Exercise 57.

$$f(x,y)=\{\begin{array}{cc}\frac{\sin \left(x{y}^3\right)}{x^2}& \text{if }x\ne 0;\\ 0& \text{if }x=0.\end{array}$$

Exercise 58.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3+y^5}{x^2+2y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 59.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2+{\sin }^2(y)}{x^2+2y^2}& \text{if }x\ne 0;\\ 0& \text{if }x=0.\end{array}$$

Exercise 60.

$$f(x,y)=\{\begin{array}{cc}\frac{\tan \left(\frac{xy}{xy+1}\right)}{xy}& \text{if }xy\ne 0;\\ 1& \text{if }xy=0.\end{array}$$

Exercise 61.

$$f(x,y)=\{\begin{array}{cc}(x^2+y^2{)}^{x^2{y}^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 1& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 62.

$$f(x,y)=\{\begin{array}{cc}\frac{(x+1)(y-x)}{(y+1)(x+y)}& \text{if }x\ne -y\text{ and }y\ne -1;\\ 0& \text{if }x=-y\text{ or }y=-1.\end{array}$$

Exercise 63.

$$f(x,y)=\{\begin{array}{cc}\frac{|y{|}^a|x{|}^b}{|x{|}^a+|y{|}^b}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 64.

$$f(x,y)=\{\begin{array}{cc}\arctan \left(\frac{y}{x}\right)& \text{if }x\ne 0;\\ 0& \text{if }x=0.\end{array}$$

Exercise 65.

$$f(x,y)=\{\begin{array}{cc}\frac{x^3{y}^3}{x^6+y^6}& \text{if }(x,y)\ne (0,0);\\ 0& \text{if }(x,y)=(0,0).\end{array}$$

Exercise 66.

$$f(x,y)=\{\begin{array}{cc}\frac{x^{3}y}{x^6+y^2}& \text{if }(x,y)\ne (0,0);\\ 0& \text{if }(x,y)=(0,0).\end{array}$$

Exercise 67.

$$f(x,y)=\{\begin{array}{cc}\frac{(x-1)y^2}{x^2+y^2}& \text{if }(x,y)\ne (0,0);\\ 0& \text{if }(x,y)=(0,0).\end{array}$$

Exercise 68.

$$f(x,y)=\{\begin{array}{cc}\frac{x^2-y^6}{x{y}^3}& \text{if }x\ne 0\text{ and }y\ne 0;\\ 0& \text{if }x=0\text{ or }y=0.\end{array}$$

Exercise 69.

$$f(x,y)=\{\begin{array}{cc}\arctan \left(\frac{2y}{x^2+y^2}\right)& \text{if }(x,y)\ne (0,0);\\ 0& \text{if }(x,y)=(0,0).\end{array}$$

Exercise 70.

$$f(x,y)=\{\begin{array}{cc}|y|\sin \left(\frac{x}{\sqrt{|y|}}\right)& \text{if }y\ne 0;\\ 0& \text{if }y=0.\end{array}$$

Exercise 71.

$$f(x,y)=\{\begin{array}{cc}{x}^2(1-\cos \left(\frac{y}{x}\right))& \text{if }x\ne 0;\\ 0& \text{if }x=0.\end{array}$$

Exercise 72.

$$f(x,y)=\{\begin{array}{cc}\sin \left(\frac{1}{xy}\right)& \text{if }x\ne 0\text{ and }y\ne 0;\\ 0& \text{if }x=0\text{ or }y=0.\end{array}$$

Exercise 73.

$$f(x,y)=\{\begin{array}{cc}\sin \left(\frac{1}{x^2+y^2}\right)& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 74.

$$f(x,y)=\{\begin{array}{cc}(x^2+y^2{)}^{\mu }\sin \left(\frac{1}{x^2+y^2}\right)& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(0,0).\end{array}$$

Exercise 75.

$$f(x,y)=\{\begin{array}{cc}\frac{\cos (xy)\sin \left(4x\sqrt{|y|}\right)}{\sqrt{|xy|}}& \text{if }x\ne 0\text{ and }y\ne 0;\\ 0& \text{if }x=0\text{ and }y=0.\end{array}$$

Exercise 76.

$$f(x,y)=\{\begin{array}{cc}{x}^n{y}^m\sin \left(\frac{1}{x^2+y^2}\right)& \text{if }(x,y)\ne (0,0);\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 77.

$$f(x,y)=\{\begin{array}{cc}\frac{1-\cos \left(x^{2}y\right)}{x^2+2y^2}& \text{if }(x,y)\ne (\mathrm{0,0});\\ 0& \text{if }(x,y)=(\mathrm{0,0}).\end{array}$$

Exercise 78.

$$f(x,y)=\{\begin{array}{cc}\frac{x^{2}y}{x^2-y^2}& \text{if }x\ne y;\\ 0& \text{if }x=y.\end{array}$$

Exercise 79.

$$f(x,y)=\{\begin{array}{cc}\frac{x-y}{x+y}& \text{if }x\ne -y;\\ 0& \text{if }x=-y.\end{array}$$