Science, Maths & Technology

### Become an OU student Metric spaces and continuity

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# Learning outcomes

After studying this course, you should be able to:

• understand the Euclidean distance function on Rn and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on Rn

• explain the definition of continuity for functions from Rn to Rm and determine whether a given function from Rn to Rm is continuous

• explain the geometric meaning of each of the metric space properties (M1) – (M3) and be able to verify whether a given distance function is a metric

• distinguish between open and closed balls in a metric space and be able to determine them for given metric spaces

• define convergence for sequences in a metric space and determine whether a given sequence in a metric space converges

• state the definition of continuity of a function between two metric spaces.
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