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Design
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Homework Statement
Prove that the distance function d : Rn x Rn -> R, defined as d(x,y) = |x-y| is continous.
The Attempt at a Solution
|x-y| >= | |x| - |y| |
|x+y| <= | |x| + |y| |
Not sure what to do from here on
thank you
Design said:So For all Epsilon>0 there exist a delta > 0
so |x-a| < delta and |y-b|<delta and | |x-y| - |a-b|| < epsilon
Where do i go from there?
When we say that distance is continuous, it means that there are no breaks or interruptions in the measurement of distance. This means that we can measure any distance, no matter how small, and it will still be a valid measurement.
While continuous distance allows for measurement at any point, discrete distance only allows for measurement at certain intervals or points. Think of it like a ruler with marks only at every inch (discrete) compared to a ruler with a continuous scale (continuous).
Continuous distance is important in science because it allows for more precise and accurate measurements. It also allows for the measurement of infinitesimally small distances, which is crucial in fields such as physics and astronomy.
In theory, distance can be discontinuous if we were to measure it at a finite level. However, in practical applications, we use continuous distance due to its precision and accuracy.
In real-life situations, we use continuous distance in various applications such as navigation, engineering, and construction. It allows for the accurate measurement of distances, whether it be for building structures or determining travel routes.