query: Proof of differentiability of $f(x)$
positive_passages.docid: 143491
positive_passages.text: Show that $f$ is differentiable on $(−1, 1)$.
negative_passages.docid: 11167
negative_passages.text: A counterexample of a differentiable function on an interval whose derivative is discontinuous
negative_passages.docid: 105084
negative_passages.text: Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
negative_passages.docid: 16418
negative_passages.text: Showing that $\frac{f(x)-f(x-h)}{h} \le f'(x) \le \frac{f(x+h)-f(x)}{h}$ given that f' is increasing
negative_passages.docid: 97382
negative_passages.text: Continuous differentiability implies Lipschitz continuity
negative_passages.docid: 150447
negative_passages.text: Prove a differentiable function between banach spaces is continuous
negative_passages.docid: 94944
negative_passages.text: Differentiable functions satisfying $f'(f(x))=f(f'(x))$
negative_passages.docid: 142870
negative_passages.text: Prove the function is not differentiable at (0,0)
negative_passages.docid: 123519
negative_passages.text: Fréchet differentiability from Gâteaux differentiability
negative_passages.docid: 4636
negative_passages.text: How to show that $f$ is differentiable at $x=0.$
negative_passages.docid: 121771
negative_passages.text: Proof $f(x)\equiv 0$
negative_passages.docid: 145173
negative_passages.text: $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$
negative_passages.docid: 56936
negative_passages.text: $\frac{\partial f_i}{x_j}=\frac{\partial f_j}{x_i}\implies(f_1,\ldots,f_n)$ is a gradient
negative_passages.docid: 68048
negative_passages.text: Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
negative_passages.docid: 80650
negative_passages.text: Complex differentiability of $f(z)=|z|$
negative_passages.docid: 117906
negative_passages.text: Show that $\liminf\limits_{x\to \infty}|f'(x)|=0$ if $\lim\limits_{x\to \infty}f(x)/x=0$ and $f$ is differentiable
negative_passages.docid: 4735
negative_passages.text: Proof for alternative definition of the derivative