Multi variable 미적분 - 편미분

■ Continuity of a function of two variables

Definition:

A function f of two variables is continuous at a point (a, b) if

The function $f$  is continuous on D if it is continuous at every point in D.

* For a function to be continuous at (a, b), it has to have a limit at (a, b) and f(a, b) should exist.  

The function f(x, y) has a domain that includes all the real numbers of x and y except the point (0, 0). So the f(a, b) is not in the f's range.  That is, f(x, y) has a hole at (0, 0), and the function is not continuous at that point.

But since f(x, y) has a limit 0 as (x, y)→(0, 0) if we define f(0, 0) to be 0, the discontinuity at (0, 0) is removed. We call such a discontinuity a removable discontinuity.  With the definition f(0, 0) = 0, the f(x, y) has become continuous on R²

Consider the function,

We saw the function does not have a limit as (x, y)→(0, 0).  So regardless of how we define f(0, 0) to be, since there isn't a limit itself as (x, y)→(0, 0), the f can't be made continuous at (0, 0) no matter what.

Properties of continuity

Like in the single variable case, if k is real number and f and g are continuous at (a, b), then the following functions are continuous at (a, b)

* It follows from the above theorem that all polynomials are continuous, and all rational functions (quotient of polynomials whose denominator is not zero polynomial) are continuous on its domain.  

Continuity of a Composite function:

If g is continuous at (a, b) and f is continuous at g(a, b), then the composite function f(g(a, b)) is continuous at (a, b).  

* The preceding definitions and theorems of the continuity can be extended to the continuity of the functions of three or more variables.

ex)  h(x, y) = g(f(x, y)) 에서 h가 어느 영역에서 연속인가?

Sol)

h(x, y) = g(f(x, y)) =

f가 polynomial 함수이므로 모든 R² 에서 연속이고 g는 t-1≥ 0 에서 연속.  

h는 2x - 3y - 6 ≥ 0 에서 연속

ex)

■ Partial Derivative of a function of two variables

z = f(x, y) 시, partial derivative 의 표현

Notation for higher-order Partial Derivatives

EQUALITY OF MIXED PARTIAL DERIVATIVES