All directional derivatives exist, yet the function is not differentiable

ProofIntermediate⏱ 30 minFoundationsdifferentiabilitydirectional derivativecounterexample

Theory: Multivariable and Matrix Calculus →

Problem

Let

f(x_1,x_2) = \begin{cases} \dfrac{x_1^3}{x_1^2+x_2^2} & (x_1,x_2)\neq(0,0),\\[2mm] 0 & (x_1,x_2)=(0,0).\end{cases}

Show that every directional derivative of $f$ at the origin exists, but $f$ is not differentiable there. Explain precisely which property of the derivative fails.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

Hints — 0 of 3 revealed

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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