The Mathematics of ECE

Real Analysis - Lecture 3

Friday, September 23, 2022

Logistics

  • Housekeeping
  • Today’s class: Real Analysis
    • Key concepts: limits, continuity, differentiability, extreme value theorem, mean value theorem, Taylor’s theorem

Limits and Continuity

  • A function \(f : \bbR \to \bbR\) is said to have a limit \(L\) as it approaches \(x_0 \in \bbR\) if \[ \forall \epsilon > 0 \exists \delta > 0 \forall x \in (\calB_\delta(x_0) \setminus x_0) :~ f(x) \in \calB_\epsilon(L) \] this is often written \(\lim_{x \to x_0} f(x) = L\).

    • Intuition: for any deviation in the codomain (\(\epsilon\)) there is a deviation in the domain (\(\delta\)) 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 \(f : \bbR \to \bbR\) is said to be continuous at a point \(x_0 \in \bbR\) if \(\lim_{x \to x_0} f(x) = f(x_0)\), i.e. \[ \forall \epsilon > 0 \exists \delta > 0 \forall x \in \calB_\delta(x_0) :~ f(x) \in \calB_\epsilon(f(x_0)) \]

    • Intuition: for any deviation in the codomain (\(\epsilon\)) there is a deviation in the domain (\(\delta\)) 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?

Mean Value Theorem

  • A continuous function \(f : \bbR \to \bbR\) 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 \(f : \bbR \to \bbR\) on the closed interval \([a, b]\) achieves its maximum and minimum values, each at least once.

  • For any differentiable function \(f : \bbR \to \bbR\) 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 \(f : \bbR \to \bbR\) 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.

Taylor’s 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 \(f : \bbR \to \bbR\) 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 \(f : \bbR \to \bbR\) be a \((k+1)\)-times differentiable function. Then there exists some \(c \in (x, x + h)\) such that \[ f(x + h) = f(x) + f'(x) h + \frac{f''(x)}{2!} h^2 + \dots + \frac{f^{(k)}(x)}{k!} h^k + \frac{f^{(k+1)}(c)}{k!} ((x + h) - c)^k h \]

Taylor’s Theorem

  • Let \(f : \bbR \to \bbR\) be a \(k\)-times differentiable function. Then there exists some \(g_k : \bbR \to \bbR\) 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 \(\lim_{h \to 0} g_k(h) = 0\)

Continuity (Revisited)

  • Let \(\calX, \calY\) be two topological spaces. A function \(f : \calX \to \calY\) is said to be continuous if \[ \forall \calV \in \tau_{\calY} :~ f^{-1}(\calV) \in \tau_{\calX} \]

    • 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.