Continuity

Question 1

When is a sequence of functions fk Riemann integrable

Answer 1

Sequence of functions fk is Riemann integrable when :
It converges uniformly to function f: [a,b] to R, and integral b down to a fk(x) dx tends to integral b down to a f(x) dx as k tends to inf

Question 2

What can lim k tends to inf integral b down to a fx(x) dx be written as

Answer 2

Lim k tends to inf integral b down to a fk(x) dx can be written as integral b down to a lim k tends to inf fk(x)dx (this = f, stuff in integral)
If have uniform convergence, can exchange integral with limit

Question 3

What is integral b down to a sum k = 1 to inf fk(x) dx

Answer 3

Integral b down to a sum k = 1 to inf fk(x) dx = sum k = 1 to inf integral b down to a fk(x) dx
Fk is sequence of Riemann integrable functions

Question 4

When is a function f continuously differentiable

Answer 4

Function is continuously differentiable when:
There is sequence of functions fk [a,b] to R where there exists x0 in (a,b) s.t sequence fk(x0) is convergent and fk’ converges uniformly on (a,b)Exists function f: (a,b) to R s.t fk tends to f and fk’ tends to f’ uniformly as k tends to inf