Continuous Vector Functions

From Department of Mathematics at UTSA
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Like with continuity of single variable real-valued functions, the continuity of vector-valued functions is as follows:

Definition: Let be a vector-valued function. We say that is Continuous at if , that is , and . We say that is Continuous on The Interval if is continuous for all .

We should note that from the definition of being continuous at , we must have that the component functions , , and all be continuous at .

For example, suppose we wanted to show that is continuous at . We notice that , , and are all continuous real-valued functions, and so they're also all continuous at . Therefore, we have that:


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