# HW 8: Challenge 3(a,b)

Consider the sequence of functions defined by

a) Prove that converges uniformly on and find .

In order to better examine this, look at , then . The limit of as is 0. This holds similarly for

For and , so converges pointwise to

Let be arbitrary and choose (which exists by Archimedean Property). Let both be arbitrary.

Examine . Notice that , since . Thus, since were arbitrary (within the stated constraints), for all such that whenever and . Thus by definition of unform converges, converges uniformly on .

b) Show that is differentiable on and compute .

As shown in part a) .

Let be arbitrary and consider . Since is constant and defined for , this is the same as . This however, if the definition of the derivative of g at c, which therefore exists. Since was arbitrary, the derivative exists for all points in , and thus by definition of differentiable, is differentiable on .

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