Limits and Continuity

Do not worry about your difficulties in mathematics. I can assure you mine are still greater, Albert Einstein

Definition. Let D ⊆ ℂ, $f: D \rarr \Complex$ be a function and let z₀ be a limit point of D. A complex number L is said to be a limit of the function f as z approaches z₀, $\lim_{z \to z_0} f(z)=L$, if for each ε > 0, there exist a corresponding δ > 0 such that |f(z) -L| < ε whenever z ∈ D and 0 < |z - z₀| < δ.

Equivalently, if for each ε > 0, there exist a corresponding δ > 0 such that whenever z ∈ D ∩ B’(z₀; δ), f(z) ∈ B(z; ε).

If no such L exists, then we say that f(z) does not have a limit a z approaches z₀.

Examples

Let $f: \overline{B(0; 1)} \rarr \Complex$

f(z) = $\begin{cases} 3z², |z| < 1 \\ 3, |z| = 1 \end{cases}$

$\lim_{z \to 1} f(z)=3$. Furthermore, $\lim_{z \to z_0} f(z)$ des not exist for any z₀ with |z₀| = 1 and z₀ ≠ ±1. 3z²➞3 if z = ±1.

However, |z| = 1, z = e^iθ, 3z² = 3e^2iθ ➞ 3 if and only if θ = π or 2π

Limit rules

If $\lim_{z \to z_0} f(x) = L_1$ and $\lim_{z \to z_0} g(x) = L_2$, then:

1. If $\lim_{z \to z_0} f(x)$ exists, then it is unique.
2. $\lim_{z \to z_0} (f(x) \plusmn g(x)) = L_1 \plusmn L2$
3. $\lim_{z \to z_0} (f(x) · g(x)) = L_1 · L2$
4. $\lim_{z \to z_0} (\frac{f(x)}{g(x)}) = \frac{L_1}{L_2}$ provided L₂ ≠ 0.
5. Since $\lim_{z \to z_0} z = z_0, ∀z_0\in\Complex$, then $\lim_{z \to z_0} f(z) = f(z_0)$ for any rational function f(z) and for every $z_0∈dom(f)$, e.g., $f(z)=\frac{(3+i)z²+2z-i}{(1+i)z⁴-z³+1}, \lim_{z \to z_0}\frac{(3+i)z_0²+2z_0-i}{(1+i)z_0⁴-z_0³+1} =f(z_0)$

Definition. Let D ⊆ ℂ, let f: D ⟶ ℂ, and z₀ be a limit point of D. We say that $\lim_{z \to z_0} f(z) = \infty$ if given M > 0, there is a δ > 0 such that ∀z ∈ D and 0 < |z -a| < δ, |f(z)| > M. Equivalently, $\lim_{z \to z_0} f(z) = \infty$ if given M > 0, there is a δ > 0 such that ∀z ∈ D ∩ B’(z₀; δ), then f(z) ∈ {w: |w| > M} -neighborhood of infinity-

Example. $\lim_{z \to 0} \frac{1}{z} = \infty$

Given M > 0, there is a δ = ¹⁄_M > 0 such that ∀z ∈ D and 0 < |z| < δ = ¹⁄_M, |¹⁄_z| > M↭ z∈D ∩ B’(0; ¹⁄_M), then ¹⁄_z ∈ {w: |w| > M}

Definition. Let D ⊆ ℂ, let f: D ⟶ ℂ. We say that $\lim_{z \to \infty} f(z) = L$ if given ε > 0, there is a M > 0 such that ∀z ∈ D, |f(z)-L| < ε wherever z ∈ {w: |w| > M}.

Example: $\lim_{z \to \infty} \frac{3z²}{(1+i)z²-z+2} = \frac{3}{1+i}$

$|\frac{3z²}{(1+i)z²-z+2} - \frac{3}{1+i}| = |\frac{3z²(1+i)-3z²(1+i)+3z-6}{(1+i)((1+i)z²-z+2)}| = |\frac{3z-6}{(1+i)((1+i)z²-z+2)}| = |\frac{\frac{3}{z}-\frac{6}{z²}}{(1+i)((1+i)-\frac{1}{z}+\frac{2}{z²})}|$ [*]

$|(1+i)-\frac{1}{z}+\frac{2}{z²})| \ge[\text{Recall: }|a-b| \ge |a| - |b|] \sqrt{2}-|\frac{1}{z}-\frac{2}{z²}|$

$|z| \ge M \leadsto \frac{1}{|z|} \le \frac{1}{M} \leadsto |\frac{1}{z}-\frac{2}{z²}| \le \frac{1}{M}+\frac{2}{M²}$

[*] $\le \frac{\frac{3}{M}+\frac{6}{M²}}{\sqrt{2}(\sqrt{2}+\frac{1}{M}+\frac{2}{M²})} = \frac{3}{\sqrt{2}}\frac{\frac{1}{M}+\frac{2}{M²}}{\sqrt{2}+\frac{1}{M}+\frac{2}{M²}} = \frac{3}{\sqrt{2}}\frac{M+2}{\sqrt{2}M²+M+2M} \le[\text{Given m large enough,} M+2 \le 2M, \sqrt{2}M²+M+2M \le 2\sqrt{2}M²] \frac{3}{\sqrt{2}}\frac{2M}{2\sqrt{2}M²} = \frac{3}{2M}$

In other words, for large M, [*] is arbitrary small, hence $\lim_{z \to \infty} \frac{3z²}{(1+i)z²-z+2} = \frac{3}{1+i}$.

Definition. Let D ⊆ ℂ. A function f: D → ℂ is said to be continuous at a point z₀ ∈ D if given ε > 0, there is a corresponding δ > 0 such that |f(z) - f(z₀)| < ε whenever z ∈ D and |z - z₀| < δ.

Alternative Definition. Let D ⊆ ℂ. A function f: D → ℂ is said to be continuous at a point z₀ ∈ D if for every sequence {z_n}^∞_n=1 such that z_n ∈ D ∀n∈ℕ & z_n → z₀, $\lim_{z_n \to z_0} f(z_n) = f(z_0)$ .

Definition. A function is said to be continuous if it is continuos at every point in its domain, e.g.,

1. Polynomial and rational functions are continuous functions.
2. If f and g are continuous at a point z₀, then f ± g, f·g are also continuous at z₀.
3. If f and g are continuous at a point z₀, g(z₀) ≠ 0, then f/g is also continuous at z₀.

Theorem. Let D ⊆ ℂ. A function f: D → ℂ is continuous if and only if the inverse image of an open set in ℂ is open in D.

Proof.

→) Suppose U ⊆ ℂ is an open set in the complex numbers. If f^-1(U) = ∅, then since ∅ is open in D, the statement holds true.

If f^-1(U) ≠ ∅, let z₀ ∈ f^-1(U) an arbitrary element, let w₀ = f(z₀). Since, by assumption U is an open set in ℂ, there is a r > 0 s.t. B(w₀; r) ⊆ U. Furthermore, since f is continuous at z₀, there is a δ > 0, s.t. |f(z)-f(z₀)| < r wherever z ∈ D and |z - z₀| < δ. In other words, f(B(z₀, δ) ∩ D) ⊆ B(w₀, r) ⊆ U, so f^-1(U) is open in D (B(z₀, δ) ∩ D is an open set)∎

←) Suppose the inverse image of an open set in ℂ is open in D. Let z₀ ∈ D and let w₀ = f(z₀). Since B(w₀; r) is an open set, there is a δ > 0 s.t., f^-1(B(w₀; r)) is open in D and since z₀ ∈ f^-1(B(w₀; r)), there is a δ > 0 s.t. D ∩ B(z₀, δ) ⊆ f^-1(B(w₀; r)) ⇒ f(D ∩ B(z₀, δ)) ⊆ B(w₀; r) ⇒ f is continuous at z₀ ∎

Definition. Let D ⊆ ℂ, let a ∈ D, B(a; r) ⊆ ℂ for some r > 0, and f: D → ℂ be a function. The derivative of f at the point a is defined to be $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ if this limits exists. If this limit exists and we express it as f’(a) or $\frac{df}{dz}|_{z=a}$, we say that f is differentiable at the point a. Otherwise, if this limit does not exist, then we say that f is not differentiable at a.

Example

Let D = ℂ - {i}, f: D → ℂ be defined as f(z) = $\frac{3z}{z-i}$. Then, f is differentiable at every point z₀ ∈ D (∀z₀ ∈ D).

Let z₀ ∈ D, $\lim_{h \to 0} \frac{f(z_0+h)-f(z_0)}{h} = \lim_{h \to 0} \frac{\frac{3(z_0+h)}{z_0+h-i}-\frac{3z_0}{z_0-i}}{h} = \lim_{h \to 0} \frac{1}{h}(\frac{3(z_0+h)(z_0-i)-3z_0(z_0+h-i)}{(z_0+h-i)(z_0-i)}) = \lim_{h \to 0} \frac{1}{h}(\frac{3z_0²+3z_0h-3iz_0-3hi-3z_0²-3z_0h+3z_0i)}{(z_0+h-i)(z_0-i)}) = \lim_{h \to 0} \frac{1}{h}(\frac{-3hi}{(z_0+h-i)(z_0-i)}) = \lim_{h \to 0} (\frac{-3i}{(z_0+h-i)(z_0-i)}) = \frac{-3i}{(z_0-i)(z_0-i)} = \frac{-3i}{(z_0-i)²}$

$f’(z_0) = \frac{-3i}{(z_0-i)²}$. Since z₀ is was taken arbitrary in the domain D, f is differentiable at every point in its domain.

Definition. A function f that is differentiable at every point in its domain is said to be differentiable, e.g., polynomial and rational functions are differentiable.