The Mathematics of ECE

Logistics

Housekeeping
- Piazza for course: sign-up
- Slides available here
Today’s class: Real Analysis
- Key concepts: limits, continuity, differentiability, extreme value theorem, mean value theorem, Taylor’s theorem

A function $f : R \to R$ is said to have a limit $L$ as it approaches $x_{0} \in R$ if $\forall ϵ > 0 \exists δ > 0 \forall x \in (B_{δ} (x_{0}) ∖ x_{0}) : f (x) \in B_{ϵ} (L)$ this is often written $lim_{x \to x_{0}} f (x) = L$ .
- Intuition: for any deviation in the codomain ( $ϵ$ ) there is a deviation in the domain ( $δ$ ) such that staying within the deviation in the domain (excepting the point itself) keeps you within the deviation in the codomain. This prevents “jumps”, but not “holes”.
A function $f : R \to R$ is said to be continuous at a point $x_{0} \in R$ if $lim_{x \to x_{0}} f (x) = f (x_{0})$ , i.e. $\forall ϵ > 0 \exists δ > 0 \forall x \in B_{δ} (x_{0}) : f (x) \in B_{ϵ} (f (x_{0}))$
- Intuition: for any deviation in the codomain ( $ϵ$ ) there is a deviation in the domain ( $δ$ ) such that staying within the deviation in the domain keeps you within the deviation in the codomain. This prevents both “jumps” and “holes”.
Drill: Is the function $f (x) = \sin (x)$ continuous?
Drill: Is the function $f (x) = | x |$ continuous?
Drill: Is the function $f (x) = lim_{a \to \infty} \exp (- a x^{2})$ continuous?

A continuous function $f : R \to R$ on the open interval $(a, b)$ is said to be differentiable if the limit $f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ exists. $f^{'} (x)$ is called the derivative at $x$ .
A continuous function $f : R \to R$ on the closed interval $[a, b]$ achieves its maximum and minimum values, each at least once.
For any differentiable function $f : R \to R$ on the closed interval $[a, b]$ such that $f (a) = f (b)$ , there exists at least one $c \in (a, b)$ such that $f^{'} (x) = 0$ .
Drill: Prove Rolle’s Theorem.
- Hint: Prove that for any differentiable function, a local minimum/maximum has derivative 0.
For any differentiable function $f : R \to R$ on the closed interval $[a, b]$ , there exists at least one $c \in (a, b)$ such that $f^{'} (c) = \frac{f (b) - f (a)}{b - a}$ .
Drill: Prove the Mean Value Theorem.

Notice that the MVT can be rearranged to yield (for $h \geq 0$ , $c \in (x, x + h)$ ) $f (x + h) = f (x) + f^{'} (c) h$
Let $f : R \to R$ be a $(k + 1)$ -times differentiable function. Then there exists some $c \in (x, x + h)$ such that $f (x + h) = f (x) + f^{'} (x) h + \frac{f^{″} (x)}{2!} h^{2} + \dots + \frac{f^{(k)} (x)}{k!} h^{k} + \frac{f^{(k + 1)} (c)}{(k + 1)!} h^{k + 1}$
Let $f : R \to R$ be a $(k + 1)$ -times differentiable function. Then there exists some $c \in (x, x + h)$ such that $f (x + h) = f (x) + f^{'} (x) h + \frac{f^{″} (x)}{2!} h^{2} + \dots + \frac{f^{(k)} (x)}{k!} h^{k} + \frac{f^{(k + 1)} (c)}{k!} ((x + h) - c)^{k} h$

Let $f : R \to R$ be a $k$ -times differentiable function. Then there exists some $g_{k} : R \to R$ such that $f (x + h) = f (x) + f^{'} (x) h + \frac{f^{″} (x)}{2!} h^{2} + \dots + \frac{f^{(k)} (x)}{k!} h^{k} + g_{k} (h) h^{k}$ and $lim_{h \to 0} g_{k} (h) = 0$

Let $X, Y$ be two topological spaces. A function $f : X \to Y$ is said to be continuous if $\forall V \in τ_{Y} : f^{- 1} (V) \in τ_{X}$
- Intuition: Similar to the real’s intuition, but replace deviations (the open balls) with any arbitrary open set.
Drill: Prove that the topological definition of continuity implies the real’s definition.
Drill: Prove that the real’s definition of continuity implies the topological defintion.