Question

Show that the function $$f(x, y)=\left\\{\begin{array}{ll} \frac{x^{2} y}{x^{6}+2 y^{2}}, & (x, y) \neq(0,0), \\ 0, & (x, y)=(0,0), \end{array}\right.$$ has a directional derivative in the direction of an arbitrary unit vector \(\Phi\) at \((0,0),\) but \(f\) is not continuous at (0,0) .

Short answer

In the given function \(f(x, y)\), we calculated the directional derivative in the direction of an arbitrary unit vector \(\Phi\) at \((0,0)\) and found that it exists and is equal to \(0\). However, when we looked at the continuity of the function at \((0,0)\), we discovered that the limit of \(f(x, y)\) as \((x, y)\) approaches \((0,0)\) is not unique and depends on the direction. Therefore, the function is not continuous at \((0,0)\). So, it's possible for a function to have a directional derivative in the direction of an arbitrary unit vector but not be continuous at a point.

Step-by-step solution

Calculate the directional derivative of \(f(x, y)\) in the direction of an arbitrary unit vector \(\Phi\) at \((0,0)\)

To find the directional derivative, we first need the gradient of the function, \(\nabla f(x,y)\). Since the function \(f(x, y)\) is defined differently for \((x, y) = (0,0)\) and \((x, y) \neq (0,0)\), we compute the gradients separately for both cases. For \((x, y) \neq (0,0)\), we find the partial derivatives of \(f(x, y)\): $$ \frac{\partial f(x, y)}{\partial x} = \frac{2xy^3 - x^5y}{(x^6+2y^2)^2} $$ $$ \frac{\partial f(x, y)}{\partial y} = \frac{x^2x^6 - 4x^6y^2}{(x^6+2y^2)^2} $$ For \((x, y) = (0,0)\), both partial derivatives are equal to \(0\). The directional derivative of \(f\) in the direction of a unit vector \(\Phi = (w, z)\) is given by the dot product of the gradient with \(\Phi\): $$ D_\Phi f(0,0) = \left(\frac{\partial f(0,0)}{\partial x},\frac{\partial f(0,0)}{\partial y}\right) \cdot (w, z) = 0 \cdot w + 0\cdot z = 0 $$ Since the directional derivative \(D_\Phi f(0, 0)\) exists and is equal to \(0\) for any unit vector \(\Phi\), we showed that the function has a directional derivative in the direction of an arbitrary unit vector \(\Phi\) at \((0,0)\).

Show that \(f(x, y)\) is not continuous at \((0,0)\)

To show that \(f(x, y)\) is not continuous at \((0,0)\), we need to find the limit of \(f(x, y)\) as \((x, y)\) approaches \((0,0)\) and compare it with the value of \(f(0,0)\). Let's find the limit by approaching an arbitrary direction: $$ \lim_{(x,y) \to (0,0)} \frac{x^2y}{x^6+2y^2} = \lim_{t\to 0} \frac{(tw)^2(tz)}{(tw)^6 + 2(tz)^2} $$ We substitute \(x = tw\) and \(y = tz\), where \(t\) is a parameter: $$ = \lim_{t\to 0} \frac{t^3w^2z}{t^6(w^6+2z^2)} = \lim_{t\to 0} \frac{t^3w^2z}{t^6(w^6+2z^2)} $$ Divide by \(t^6\): $$ = \lim_{t\to 0} \frac{w^2z}{w^6+2z^2} $$ The limit exists and is equal to \(\frac{w^2z}{w^6+2z^2}\), which depends on the direction \((w, z)\). Thus, the limit is not unique, so the function is not continuous at \((0,0)\). In conclusion, the function \(f(x, y)\) has a directional derivative in the direction of an arbitrary unit vector \(\Phi\) at \((0,0)\), but it is not continuous at \((0,0)\).

Key concepts

Continuity in Multivariable Calculus

Continuity in multivariable calculus is about understanding how a function behaves as its input values approach a certain point. In single-variable calculus, we know a function is continuous at a point if the limit exists and equals the function's value at that point. This idea extends to functions with multiple variables.
In the context of multivariable functions, such as those with inputs \((x, y)\), continuity means that as the point \((x, y)\) approaches \((a, b)\), the value of the function approaches \(f(a, b)\). For this to hold, the limit must be the same no matter the path taken to get to \((a, b)\).
For example, if you approach the origin \((0,0)\) along the path \((x, 0)\) or \((0, y)\), the results should be the same to maintain continuity. If different paths lead to different limits, the function is not continuous at that point.
Discontinuity in a point means the function can behave unexpectedly, posing challenges when trying to apply calculus tools like derivatives and integrals. That's what makes continuity so crucial in the analysis of multivariable functions.

Gradient of a Function

The gradient of a function, often denoted as \(abla f\), is a vector that contains all the partial derivatives of the function. It points in the direction of the steepest increase of the function, and its length indicates how steep this increase is.
For example, consider a function \(f(x, y)\). Its gradient at any point is given by \(abla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\). This vector not only provides direction but also helps determine the slope along any path.
The gradient is crucial for finding directional derivatives, which show how the function changes in any given direction. By taking the dot product of the gradient vector and another direction vector, you can compute this directional derivative and understand how fast the function is changing in that specific direction.

• The gradient is zero at maximum, minimum, and saddle points, where the function's slope is flat.
• In applications, gradients are used to optimize functions and model physical processes involving waves or heat.

Limit in Multivariable Calculus

Understanding limits in multivariable calculus is crucial for mastering continuity and derivatives. Unlike single-variable calculus, where you approach a point along a straight line from two directions, multivariable limits can be approached from infinitely many paths.
The limit of a function \(f(x, y)\) as \((x, y)\) approaches \((a, b)\) is heading towards a singular value, but only if every possible path to \((a, b)\) leads to the same result.
For instance, if you evaluate a function as \((x, y)\) approaches \((0,0)\) and you find that the limit depends on which path you take, the overall limit does not exist. This is because a single multivariable limit must be path-independent.

• If a limit is path-dependent, the function cannot be continuous at that point.
• Different paths revealing different limits demonstrate that the behavior of the function is inconsistent and unpredictable at that point, justifying its discontinuity.