Graph of Continuous Real Function is Manifold


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Theorem

Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space.

Let $f : U \to \R^k$ be a continuous mapping.

Let $\map \Gamma f$ be the graph of $f$ equipped with the subspace topology.


Then $\map \Gamma f$ is a $n$-manifold.


Proof

Let $\gamma_f : U \to \R^{n + k}$ be the graph parametrization of $\map \Gamma f$.

From Graph Parametrization of Continuous Mapping is Embedding, $\gamma_f$ is an embedding.

The image of $\gamma_f$ is $\map \Gamma f$.

Therefore, $U$ and $\map \Gamma f$ are homeomorphic.

In other words, for each point in $\map \Gamma f$, $\map \Gamma f$ is a open neighborhood that is homeomorphic to an open subset of $\R^n$.

So $\map \Gamma f$ is locally Euclidean of dimension $n$ by definition.

From Locally Euclidean Subspace of Euclidean Space is Manifold, $\map \Gamma f$ is a $n$-manifold.

$\blacksquare$


Sources

  • 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings
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