Let \(f:\mathbb R^n\to\mathbb R\) satisfy the following conditions: \(f\) is differentiable on \(\mathbb R^n\setminus\{0\}\), \(f\) is continuous at \(0\), and \[ \lim_{p\to0}\frac{\partial f}{\partial x_i}(p)=0 \] for \(i=1,\ldots,n\). Prove that \(f\) is differentiable at \(0\).
Solution
Use the Euclidean norm on \(\RR^n\).
1. For every \(\varepsilon>0\) there is \(\delta>0\) such that\[ 0<\norm p<\delta \quad\Longrightarrow\quad \abs{\frac{\partial f}{\partial x_i}(p)} < \frac{\varepsilon}{\sqrt n} \]for every \(i=1,\ldots,n\).
For each \(i\), the hypothesis \[ \lim_{p\to0}\frac{\partial f}{\partial x_i}(p)=0 \] gives a radius \(\delta_i>0\) on which the displayed bound holds for the \(i\)th partial derivative. Since there are only finitely many indices, take \[ \delta=\min_{1\leq i\leq n}\delta_i. \]
2. If \(0<\norm x<\delta\), then\[ \abs{f(x)-f(0)} < \varepsilon\norm x. \]
Fix such an \(x\) and define \[ g(t)=f(tx), \qquad 0\leq t\leq1. \] The continuity of \(f\) at \(0\) makes \(g\) continuous at \(t=0\). For \(0<t\leq1\), the point \(tx\) is nonzero, so differentiability of \(f\) at \(tx\) implies continuity there; hence \(g\) is continuous on \([0,1]\). The chain rule also shows that \(g\) is differentiable for \(0<t<1\). The one-variable mean value theorem therefore gives \(c\in(0,1)\) such that \[ f(x)-f(0) = g'(c) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(cx)x_i. \] Because \[ 0<\norm{cx}<\delta, \] step 1 yields \[ \begin{aligned} \abs{f(x)-f(0)} &\leq \frac{\varepsilon}{\sqrt n} \sum_{i=1}^n\abs{x_i}\\ &\leq \frac{\varepsilon}{\sqrt n} \sqrt n\norm x\\ &= \varepsilon\norm x, \end{aligned} \] where the second inequality is Cauchy–Schwarz.
3. The function \(f\) is differentiable at \(0\) with derivative thezero linear map.
Step 2 shows that for every \(\varepsilon>0\), \[ 0<\norm x<\delta \quad\Longrightarrow\quad \frac{\abs{f(x)-f(0)}}{\norm x}<\varepsilon. \] Hence \[ \lim_{x\to0} \frac{\abs{f(x)-f(0)}}{\norm x} = 0. \] This is precisely differentiability at \(0\) with derivative equal to the zero linear map.
4. Q.E.D.
Step 3 is the required conclusion.