continuity of f(x,y)
a function f(x,y) is continuous at (a,b) if limf(x,y)=f(a,b) as (x,y) →(a,b)
To determine the continuity of a function f(x,y), we need to check three conditions:
1. Existence of limit: We first need to check if the limit of f(x,y) exists as (x,y) approaches a given point (a,b). This means that f(x,y) has to approach a single finite value as (x,y) gets closer to (a,b).
2. Functional value: The functional value of f(x,y) at the point (a,b) must exist, meaning that f(a,b) should be defined.
3. Value of limit: The value of the limit of f(x,y) as (x,y) approaches (a,b) must be equal to the functional value of f(x,y) at (a,b).
If all three conditions are satisfied, then f(x,y) is continuous at (a,b). If any one of the conditions fails, then f(x,y) is not continuous at (a,b).
It’s important to remember that if a function is defined piecewise, then we need to check the continuity at each of the points where the function changes.
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