The function f: ℝ² → ℝ defined as f(x,y) = (x³ - y³) / (x - y) for x ≠ y can be continuously extended to the line y = x by defining f(x, x) = 3x². This definition ensures continuity at the line y = x, as the limit of f(x, y) approaches 3x² when x approaches y. The misconception that f(x, x) should equal 0 is incorrect; the correct value is derived from the polynomial simplification of the function.
PREREQUISITESStudents and educators in mathematics, particularly those focusing on calculus and real analysis, as well as anyone interested in understanding the continuity of functions in multiple dimensions.
<br /> <br /> Thank you very much!HallsofIvy said:No. Did you have any reason at all for thinking that?
For x- y\ne 0
\frac{x^3- y^3}{x- y}= x^2+ xy+ y^2[/itex]<br /> <br /> As long as x is NOT equal to y, that function is the same as x^2+ xy+ y^2. The function will be continuous on the line the line x= y, if we define f(x, y)= f(x, x)= x^2+ xy+ y^2= x^2+ x^2+ x^2= 3x^2, not "0".