262CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATIONWe will exploit this equation in two ways. For notational convenience, wewill drop reference to where a partial is being taken:for the rest of thisproof,∂f∂x=(x,y)∂y=()where−→x= () is the point at which we are trying to provedi±erentiability ofφ.Moving the ²rst two terms to the left side, dividing by (△x)(∂f∂y), andtaking absolute values, we havevvvv△y△x+∂f/∂x∂f/∂yvvvv=|ε|||b(△x,△y)b|△x|≤|ε|||b1 +vvvv△y△xvvvvB(3.20)(sinceb(△x,△y)b ≤ |△x|+|△y|). To complete the proof, we need to ²ndan upper bound forvvv1 +△y△xvvvon the right side.To this end, we come back to Equation (3.19), this time moving just thesecond term to the left, and then take absolute values, using the triangle
This is the end of the preview.
access the rest of the document.