Operations on the continuity of a function of two variables

 Operations on continuity of a function of two variables:

Theorem 1

The sum of continuous functions is continuous

If a function f(x,y) is continuous at (x₀.y₀), then f(x,y) + g(x,y) is continuous at (x₀.y₀). 

Theorem 2

The product of continuous functions is continuous

If g(x) is continuous at x₀ and h(y) is continuous at y₀, then f(x, y) = g(x).h(y) is continuous at (x₀.y₀).

Theorem 3

The composition of continuous functions is continuous

Let g be a function of two variables from

Let's suppose g continuous at  (x₀.y₀) and let z₀ = g(x₀.y₀). Let f be a function that relates elements from R to R such that z₀ is in the domain of f. Then 

Example

Show that f(x,y) = 4x³y² and g(x,y) = cos(4x³y²) are continuous everywhere

Solution

a) Continuity of f(x.y) = 4x³y²

We have f(x) = 4x³ is continuous for every real number; f(y) = y² is continuous for every real number.. Therefore f(x,y) = f(x).f(y) as product of two continuous functions is continuous for every real number.

b) Continuity of g(x, y) =  cos(4x³y²) 

We have f(x, y) =4x³y² which is continuous for every element of  R²; h(x) = cosx is continuous for every element of R . Let Let f(x,y) = z. We have h(z) = cosz ; h[f(x,y)] = cos[f(x, y)] ; foh[(x,y)] = cos (4x³y²) is continuous for every element of R² as a composition of 2 functions. We can see that foh(x,y) is nothing more than the function f(x,y), Therefore  f(x.y) = 4x³y² is continuous for every element of R².

Practice

Show that the functions f(x, y) = 2x²y³+ 3 and g(x,y) =  2x²y³+ 3)⁴ are continuous everywhere.