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# Math HW help with functions

**Sequences of Functions**

**Def**. Let (*fn*) be a sequence of functions defined on a subset *S* of **R**. Then (*fn*) **converges uniformly** on *S* to a function *f* defined on *S* if

For each e > 0 there exists a number *N* such that for all *x* in* S*, for all *n* > *N* , |* fn* (*x*) -*f*(*x*) | < e.

**#4**.Let for *x* Ã Let *f *(*x*) = 0.

Complete the following discussion and proof that (*fn*) converges uniformly to *f* on .

**Discussion:**

Suppose e is any positive real number.

We want to find *N* such that for all *x* Ã and *n* > *N*, we have |*fn*(*x*) -*f* (*x*)| =< e

Note that since x Â³ we haveand Â£ **____**.(** ____are numbers in simplest form)**

**______**for all *x* Ã

(**___ is an expression involving an appropriate constant and the variable n only, no x)**

So, we want**______**< e, which implies that *n* > _____.

**Proof:**

Let > 0. Choose *N* = _____. For all *x* Ã , and *n* > *N*, we have

**______** <**_______**= e, as desired.

**(______ **should be the expression involvinge, before being simplified to get exactly e.)

**#5**.Let for *x* Ã **R**.Let *f*(*x*) = 3*x*2.

Clearly, (*fn*) converges pointwise to *f*. But does it converge uniformly to *f*?

Fill in the blanks to carefully show that (*fn*) does ** not **converge uniformly to

*f*on

**R**.

We must show: (the negation of the definition)

For _____ (all/some) e > 0, for _____ (all/some) *N, *for_____ (all/some) *x* in**R**and _____ (all/some) *n*> *N* ,

|* fn* (*x*) - *f*(*x*) | __(<,>,Â£, Â³)e.

Let = 1. Given any *N* , let *n* be a positive integer greater than *N*, and set*x* = e*n*.

Then we have |* fn* (*x*) - *f*(*x*) | =______________________________________ (<,>,Â£, Â³)1 = e.

(NOTE: In the _____________________substitute for*fn* (*x*) and *f*(*x*) and simplify, applying *x* = e*n**.*)

**#6**.Let for *x* Ã [0, 1].

**#6(a) **State *f *(*x*) = lim *fn*(*x*).

**#6** **(b) **Determine whether (*fn*) converges uniformly to *f* on[0, 1].Carefully justify your answer, either showing that the definition (shown on the previous page) holds, or proving that it cannot hold.

**#7**.Letfor *x* Ã [-0.8, 1].

**#7** **(a)** State a formula for *f *(*x*) = lim *fn*(*x*).(no explanation required)

**#7** **(b)**(*fn*) does not converge uniformly to *f* on [-0.8,1].How can you deduce this fact without working hard? Please explain. HINT: Apply an appropriate theorem and an observation about the function* f*(*x*) you found in part (a).

**Series of Functions (#8, 12 pts)**

**#8**. Determine whether or not the given series of functions converges *uniformly* on the indicated set. Justify your answers with work/explanations . (HINT: Consider the Weierstrass M-Test and find an appropriate sequence *Mn* )

**#8 (a)**for *x* in **R**.

**#8(b)**for *x* Ã.

**#9.**Show that it is possible to have a sequence of functions that are discontinuous at every point yet converge uniformly to a function that is continuous everywhere.

That is, **state an example of a sequenceof functions ( fn)and a function f satisfying all of the following**:

- Each
*fn*is discontinuous at every real number. - (
*fn*)converges uniformly to*f*. *f*is continuous at every real number.

(explanation not required)