Question

Consider d on R defined by

d(x, y) = ?|x − y|.

(1) Show that (R, d) is a metric space.

(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite length.

Remark: On (2), you only need to verify by the partitions of equal distances. Although this is slightly different from the actual definition, it indeed implies that length equals to infinity, by using some techniques in the Riemann sum (e.g. refining a partition). This is not an analysis course, so you only need to verify with partitions of equal distances. Also, (2) implies that (R, d) is not a length metric.

3. Consider d on R defined by d(x, y) = (x – yl. (1) Show that (R, d) is a metric space. (2) Show that the path y(t) = t.te [0, 1] has infinite length. Remark: On (2), you only need to verify by the partitions of equal distances. Although this is slightly different from the actual definition, it indeed implies that length equals to infinity, by using some techniques in the Riemann sum (e.g. refining a partition). This is not an analysis course, so you only need to verify with partitions of equal distances. Also, (2) implies that (R, d) is not a length metric.

Answer 1

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