Is (x^2+y^2)sin(1/(x^2+y^2)) differentiable? Real Analysis II

In this lesson, we worked through four key exercises to analyze the scalar-valued piecewise-defined function: f(x,y) = (x^2+y^2)sin(1/(x^2+y^2)) when (x,y) is not (0,0), 0 at (x,y) = (0,0). We show that the function is continuous at the origin by calculating the limit of f(x, y) as (x, y) approaches the origin. Using the squeeze theorem, we demonstrated that the product of the bounded sine term and the quadratic term goes to zero, matching the value of the function at the origin. Next, we computed the partial derivatives of f with respect to x and y at the origin using the limit definition. This allows us to construct the Jacobian matrix at the origin, even before we know that the function is differentiable--this is something to keep in mind about functions of multiple variables (existence of partials does not imply differentiability). Next we examine the general expressions for df/dx and df/dy as piecewise-defined functions. Away from the origin, the derivatives were computed directly using standard techniques. At the origin, the derivatives were defined to be zero. We consider the theorem seen in an earlier video: if the partials are continuous on an open neighborhood of (0,0), then the function is differentiable there. Does that apply here?... (Theorem here: https://youtu.be/kWDxQ1VS-XQ) Finally, we used the formal definition of differentiability to assess whether f is differentiable at the origin. This function exemplifies a surprising result in calculus: a function can be differentiable even if its partial derivatives are not continuous. To better understand this function’s behavior, we compared it to its 1D analog, x^2 * sin(1 / x^2), which exhibits similar oscillatory behavior near the origin. This analogy helps visualize the rotational symmetry of the 2D function and understand its differentiability. #Realanalysis #advancedcalculus #MultivariableCalculus #Differentiability #PartialDerivatives #jacobianmatrix #differentiation #maths #math