MHB Quick question about continuous mapping

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A function f mapping a subset E of a metric space X into another metric space Y is continuous if it is continuous at every point p in E. This means that the definition of continuity at a point aligns with the broader definition of continuity for the entire function. Therefore, stating that f is a continuous mapping is equivalent to asserting that it is continuous at each point in E. This understanding is fundamental in topology and metric space analysis. Continuity at individual points ensures overall continuity of the function.
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When f maps E into a metric space Y: (E is subset of metric space X)
Is it eqivalent to say that f is a continuous mapping and that for a subset E of X, to say that for every p element of E, f is continuous at p.?

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Yes. To say that a function (or mapping) is continuous is the same as saying that it is continuous at each point of its domain.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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