hedipaldi said:convert to ellyptic coordinates x=2rcos(t); y=sqt(2)rsin(t) and compute the limit as r tends to 0,using L'hopital's rule.If the limit is not identically 0 (or depends on t),then the function is not continuous at the poins on the ellipse.
Continuity of a multivariable function means that the function remains unchanged at a given point when the input variables are changed by a small amount. In other words, the limit of the function as the input variables approach a point must exist and be equal to the value of the function at that point.
In a single variable function, continuity is determined by the behavior of the function at a single point. However, in a multivariable function, continuity is determined by the behavior of the function at every point in the domain. This means that all possible paths to a point must produce the same limit for the function to be continuous.
Yes, it is possible for a multivariable function to be continuous at a point but not differentiable. This can occur when the function has sharp turns or corners at that point. In order for a function to be differentiable, it must also be smooth and have a well-defined tangent plane at that point.
No, not all continuous multivariable functions are differentiable. A function may be continuous at a point but not differentiable if it has sharp turns or corners at that point. However, if a function is differentiable, it must also be continuous.
The continuity of a multivariable function can be tested using the three-variable limit theorem. This theorem states that if a function is continuous at a point, the limit of the function as three variables approach that point must be equal to the value of the function at that point. Additionally, the continuity of a multivariable function can also be checked by evaluating the function along different paths to the point in question and ensuring that the limit is the same for all paths.