3.4. LEVEL CURVES261theng′a(y) =∂f∂y(a,y)>0soga(y) is strictly increasing on [y−δ2,y+δ2]. In particular, whena=x0,gx0(y0−δ2)<0<gx0(y0+δ2)and we can pickδ1>0 (δ1≤δ) so thatga(y0−δ2)<0<ga(y0+δ2)for eacha∈[x0−δ1,x0+δ1]. The Intermediate Value Theorem insuresthat for each suchathere isat least oney∈[y0−δ2,y0+δ2] for whichga(y) =f(a,y) = 0, and the fact thatga(y) is strictly increasing insuresthat there isprecisely one. Writingxin place ofa, we see that thedefinitionφ(x) =y⇐⇒f(a,y) = 0and|y−y0|<δ2gives a well-defined functionφ(x) on [x0−δ1,x0+δ1] satisfyingEquation (3.17).Secondwe show that this function satisfies Equation (3.18).We fix(x,y) = (x,φ(x))in our rectangle and consider another point
This is the end of the preview.
access the rest of the document.