Limits and continuity

Question 1

f(z) tends to l as z tends to zo

Answer 1

Suppose that f is defined in a set S C_ (C for which zo is an interior point or boundary point.
We say f(z) tends to l as z tends to zo if for every e>0 there exists a d>0 such that if 0 < |z-zo|< d then |f(z) -l| < e.

Question 2

Triangle inequality for complex numbers

Answer 2

|z| - |zo| | =< | z - zo |

Question 3

Limit exists and is unique

Answer 3

Suppose that zo is an interior or boundary point of the set S C_ (C, and that f : S -> (C is a function. If the limit (z-> zo, z e S) f(z) exists, then it is unique

Question 4

Rewriting limits

Answer 4

Suppose that zo is an interior or boundary point of the set S C_ (C, and that f: S -> (C is a function. Then
lim (z-> zo) f(z) = lim (w -> 0) f(zo + w).
in the sense that if one of these limits exists then so does the other, and they are equal.

Question 5

f(z) tends to l as z tends to infinity

Answer 5

Suppose that the function f is defined in an unbounded set S C_ (C and that l e (C. Then we say f(z) tends to l as z tends to infinity in S, if for every e>0, there exists r>0 such that
| f(z) - l | < e when |z| > r and z e S.

Question 6

f(z) tends to infinity as z tends to zo

Answer 6

Suppose that f is defined in a set S C_ (C for which zo is an interior point or a boundary point. We say that f(z) tends to infinity as z tends to zo if for every r e R+, there exists a d>0 such that
| f(z) | > r when 0< |z - zo| < d and z e S.

Question 7

Algebra of complex limits

Answer 7

Suppose that S C_ (C, that either zo is an interior point or a boundary point of S, or S is unbounded and zo = infinity, that c e (C, and that f : S -> (C and g: S -> (C are functions.
Then
(1) lim (z->zo) cf(z) = lim (z->zo) f(z)
(2) lim (z->zo) f(z) + g(z) = lim (z->zo) f(z) + lim (z->zo) g(z)
(3) lim (z->zo) f(z)g(z) = lim (z->zo)f(z) lim (z->zo)g(z)
(4) lim (z->zo) f(z)/g(z) = lim (z->zo)f(z) / lim (z->zo) g(z)

Question 8

Limit of a complex conjugate

Answer 8

Suppose that S C_ (C, that either zo is an interior or a boundary point of S, or S is unbounded and zo = infinity, and that f : S -> (C is a function. Then
_____ _________
lim (z->zo) = lim (z->zo) f(z)

Question 9

Limit of the real and imaginary parts

Answer 9

(1) lim (z->zo) Re (f(z)) = Re ( lim (z->zo) f(z) )
(2) lim (z->zo) Im (f(z)) = Im ( lim (z->zo) f(z) )
(3) lim lim (z->zo) f(z) = lim (z->zo) Re ( f(z) ) + i lim (z->zo) Im (f (z))

In particular, f(z) tends to l iff Re (f(z)) tends to Re(l) and Im ( f(z)) tends to Im ( l )

Question 10

Continuity

Answer 10

Suppose that the function f is defined in a set S C_ (C, and that zo e S. We say that f is continuous at zo if
lim ( z -> z0) f(z) = f(zo).

We say that f is continuous in S if it is continuous at all points of S.

Question 11

Continuity of composition of functions

Answer 11

Suppose that f : S -> (C and g : T -> (C are continuous functions in S C_ C and T C_ (C. Then
f.g is continuous in {z e T : g(z) e S}

Question 12

Maximum of a compact set

Answer 12

Suppose that the set S C_ (C is compact and that f is a continuous function defined on S. Then there exists a point zo in S such that
| f(zo) | = max { |f(z)| : z e S}