How can you tell if fxy is continuous?

Continuity

1. f is continuous at (x0,y0) if lim(x,y)→(x0,y0)f(x,y)=f(x0,y0).
2. f is continuous on B if f is continuous at all points in B. If f is continuous at all points in R2, we say that f is continuous everywhere.

Is x 2 y continuous?

y = x2 is continuous at x = 4. In the function g(x), however, the limit of g(x) as x approaches c does not exist. If the left-hand limit were the value g(c), the right-hand limit would not be g(c).

Is xy continuous?

Example. The functions sin(xy), x^2y^3+ln(x+y), and exp(3xy) are all continuous functions on the xy-plane, whereas the function 1/xy is continuous everywhere except the point (0,0).

How do you show that a function is continuous at 0 0?

To prove that f is continuous at 0, we note that if 0 ≤ x<δ where δ = ϵ2 > 0, then |f(x) − f(0)| = √ x < ϵ. f(x) = ( 1/x if x ̸= 0, 0 if x = 0, is not continuous at 0 since limx→0 f(x) does not exist (see Example 2.7).

Is the function differentiable at 0 0 )? Demonstrate why or why not?

Zero is particularly convenient because in general, when both partial derivatives are 0, then any directional derivative must also be 0 unless f was not differentiable at that point. Thus in order to show that it is not differentiable at (0,0) it suffices to show a linear path that leads to a different derivative.

Is zero a continuous function?

f(x)=0 is a continuous function because it is an unbroken line, without holes or jumps. All numbers are constants, so yes, 0 would be a constant.

What is the key difference between limit and continuity?

A limit is a certain value. Continuity describes the behavior of a function. In calculus, a limit is the first thing you learn, and it is the value that a function of x approaches as its x-value approaches a certain value.

Is $F(0/0)$ a continuous function?

Now, notice that in your case $f(0,0)$ is undefined, so that this statement doesn’t make sense. Every time a function is undefined in some point it cannot be continuous there because this kind of statement won’t make sense.

How do you know if a function is continuous?

I understand that a function is continuous if limit as $(x,y)$ approaches $(a,b)$ of $f(x,y) = f(a,b)$ but I am still a bit confused as to the answer.

Can a function be continuous if it is undefined?

Every time a function is undefined in some point it cannot be continuous there because this kind of statement won’t make sense. In other words, (0, 0) ∉ A and it’s clear from the definition that a function can only be continuous at a point of it’s domain.

How to prove that f is continuous in set D?

Second question: Second, I computed f x = y ( x 2 + y 2) − x y ( 2 x) x 2 + y 2, f y = x ( x 2 + y 2) − x y ( 2 y) x 2 + y 2. How to proof f was continuous through the Corollary that: if all the first partial derivatives of f exists and are continuous in an open set D, then f itself is continuous in D.

https://www.youtube.com/watch?v=sj6qPJD1WDE