MA101.17 Continuity of functions of two variables

We can say that f(x,y)\to L as the (x,y) \to (a,b), or \displaystyle \lim_{(x,y)\to (a,b)} f(x,y) = L.

If, intuitively speaking, by going close enough to (a,b) we can get f(a,b) as close as we like to L.

f(x,y) is said to be continuous at (a,b) if \displaystyle \lim_{(x,y)\to (a,b)} f(x,y) = f(a,b).

Just as with one variable limits from below and above may be different, with two variables we may get various limits coming in from different paths.

Example

\displaystyle f(x,y)=\frac{xy}{x^2+y^2} \qquad (x,y) \neq (0,0)

Consider coming in to (0,0) along the line y=mx (m fixed). Along this line we have:

\displaystyle f(x,y) = f(x,mx) = \frac{xmx}{x^2 + m^2x^2} = \frac{m}{1+m^2}

Thus along y=mx,

\displaystyle \lim_{(x,y)\to (0,0)} f(x,y) = \lim_{x\to 0} \frac{m}{1+m^2} = \frac{m}{1+m^2}

If we come in along the line y=x^2,

\displaystyle \lim_{along y=x^2} f(x,y) = \lim_{x\to 0} \frac{x^3}{x^2+x^4} = \lim_{x\to 0}\frac{1}{\frac{1}{x}+x} = 0.

MA101.9 Continuity

Let a be a point in the domain of a function. The function is said to be continuous at x=a if \displaystyle\lim_{x\to a^+} f(x)= \lim_{x\to a^-} f(x) = f(a).

Otherwise f is said to be discontinuous at x=a. The function may be discontinuous at a for various reasons:

i) \displaystyle\lim_{x\to a^-} f(x) may not exist, \displaystyle\lim_{x\to a^+} f(x) may not exist, or both may not exist.
ii) both may exist, but may not be equal.
iii) both may exist, and be equal, but not be equal to f(a).

If f is continuous at all points of its domain it is simply said to be continuous. (Sometimes for emphasis mathematicians say f is continuous everywhere).

Note that it is a silly question to ask whether or not a function is continuous at a point which is not in the domain. For example is \frac{x(x-1)}{x-1} continuous at x=1?