Find examples of functions f : R2 -> R and g : R2 -> R with the following proper

Question

Find examples of functions f : R2 -> R and g : R2 -> R with the following properties:

(a) f is continuous on the region R^2 {(x, y) is an element of R^2 | x = 0 or y = 0} , but not continuous on { (x,y) included {(x,y) is an element of R^2 | x=0 or y=0 }

(b) g is continuous on the region R^2 {(x,y) is an element of R^2 | y=0 and x is a an element of Natural number}, but not continuous on {(x,y) is an element of R^2 | y=0 and x is an element of Natural number}.

Provide an argument for your claim, and describe in both cases the region of continuity.

Explanation / Answer

There are some theorems which are very useful in cases like this, like x mapsto x is continuous a scalar multiple of a continuous function, as well as the sum of two continuous functions is continuous a product of two continuous functions is continuous a quotient of two continuous functions is continuous as long as the denominator is non-zero the composition of continuous functions is continuous, where defined There should be such a set of theorems which is proved almost right after the definition of continuous, which we usually apply instead of the definition. If you want to do the definition, you should calculate the limit, for example lim_{(x, y) o (a, b)} (3 y^2 + x^3) and show that it is 3b^2 + a^3.

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