Please show that the graph of $f(x)=|x|^λ$, where $k<λ<k+1$ is a differentiable manifold of class $C^k$ but not class $C^{k+1}$. (Problem in Differential Topology, Hirsch, Morris W.) Or, please show other examples.
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1Which part confuses you? Why the graph of a $C^k$ map is a $C^k$-submanifold, or why that function is a $C^k$ map for $\lambda <k<\lambda +1$? Or maybe both? – Sandesh Jr Feb 22 '19 at 13:52
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1I cannot understand the reason why the graph of $f(x)=|x|^\lambda$ is not a $C^{k+1}$-submanifold. – Adele Feb 26 '19 at 01:21