Question 4

Question 1

Continuous:

Answer 1

A function f:X->R is continuous at ceX iff for all E>0 there exists delta>0 s.t. for all xeX, if /x-c/<delta, then /f(x)-f(c)/<E.

Question 2

Discontinuous:

Answer 2

A function f:X->R is called discontinuous at ceX iff it is not continuous at c. There exists E>0 s.t. for all delta>0 there exists xeX, with /x-c/<delta, and /f(x)-f(c)/≥E.

Question 3

Sequentially Continuous:

Answer 3

A function f:X->R is sequentially continuous at ceX iff, for any sequence (Xn)cX that converges to c, the sequence (f(Xn)) converges to f(c).

Question 4

Sequentially discontinuous:

Answer 4

A function f:X->R is sequentially discontinuous at ceX iff, there exists a sequence (Xn)cX that converges to c, s.t. the sequence (f(Xn)) does not converge to f(c).

Question 5

How do we identify continuity and sequential continuity together?

Answer 5

A function f: X -> R is continuous at c e X if and only if it is sequentially continuous at c.

Question 6

Sandwich theorem:

Answer 6

f,g,h,: X->R, then f(x)≤g(x)≤h(x) for all xeX. If the limit of f(x)=limit of h(x)=v from x->c, then the limit of g(x)=v from x->c.

Question 7

Intermediate value theorem:

Answer 7

Suppose that f:[a,b] -> R is continuous and that f(a)<f(b). Then for all ve(f(a),f(b)), there exists ce(a,b) s.t. f(c)=v