Let f be the function in Exercise 49 . Now let (x, y)...

Let $f$ be the function in Exercise $49 .$ Now let $(x, y)$ approach (0,0) along the curves $y=m x^{2}$ and see what happens to $f(x, y)$. That is, find $\lim _{x \rightarrow 0} f\left(x, m x^{2}\right)$. Do you obtain the same limit for every value of $m ?$ What do you now conclude about the continuity of $f$ at (0,0)$?$

Key Concepts

Continuity of Multivariable Functions

For a function to be continuous at a point in multiple dimensions, the limit of the function as the variables approach that point must exist and equal the function's value at that point. If the limit is path-dependent or does not match the function's value, then the function is not continuous at that point.

Substitution via Parameterized Curves

This technique involves substituting a specific relationship between the variables (such as y = m*x^2) to examine how the function behaves along a particular path. By analyzing the limit along these curves, one can determine if the function's limit depends on the path taken, which is key to understanding the overall behavior and continuity of the function.

Multivariable Limits

This concept deals with taking limits for functions with more than one variable. The behavior of a function is studied as the input variables approach a particular point from various directions. Unlike single-variable limits, in multiple dimensions, the limit must be unique along all possible paths for the limit to exist.

Path Dependence of Limits

In the context of multivariable calculus, a limit is said to be path-dependent if the value approached by the function differs when taken along different paths to the same point. This indicates that the overall limit does not exist because the function does not settle on a unique value.

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