Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Analysis & Linear Algebra › Functions of Several Variables

“If all partial derivatives exist at a point then f is continuous there” — false

Counterexample: f(x,y) = xy/(x²+y²), f(0,0) = 0

Both partials are 0 at the origin, but f = ½ along y = x, so f is not continuous.

multivariable

#2 · Analysis & Linear Algebra › Functions of Several Variables

“Mixed partial derivatives are always equal” — false

Counterexample locked — unlock with Notes + PYQ

multivariable

#3 · Analysis & Linear Algebra › Functions of Several Variables

“A C¹ map with everywhere non-zero Jacobian is injective” — false

Counterexample: f(x,y) = (eˣ cos y, eˣ sin y) on ℝ²

The Jacobian determinant is $e^{2 x}$ ≠ 0, but $f (x, y) = f (x, y + 2 π)$ . Invertibility is only local.

multivariable