Pure Mathematics D4: Continuity

15 flashcards · Mathematics · by lakesare

  1. What exactly do we mean by "function f is defined on a set I"?

    f is defined on I ≡ I ⊆ domain(f)

  2. [Notation] What does this mean in a more usual notation?

  3. [Creating new functions] How do we find the domain for each of these new functions: for f + g, fg, λf, f/g for g ◦ f for f –1

    for f + g, fg, λf, f/g – very heuristic domain domain(f) ∩ domain(g) - { undefined } for g ◦ f – whatever is comfortable for both functions for f –1 – range of the function f

  4. How can you obtain the graph of f–1 from the graph of f?

    Reflect f across the y = x line.

  5. Name 3 ways to determine whether f is invertible. A function is invertible iff it's one-to-one.

    Original definition: show that f(x1) = f(x2) → x1 = x2 Explicit rule-finding: determine f–1 by finding the rule Unusual: show that f is strictly increasing or decreasing

  6. Monotonic VS strictly monotonic function?

    monotonic – increasing or decreasing strictly monotonic – strictly increasing or strictly decreasing

  7. Visually imagine 2 ways to think of sequences: if lim n → ∞ xn = 2 then lim n → ∞ 3xn = 6 lim x → 2 3x = 6 Function f is continuous at some point c of its domain if the limit of f(x), as x approaches c through the domain of f, exists and is equal to f(c).

  8. Express continuity definition as a limit.

    Function f is continuous at some point c of its domain if the limit of f(x), as x approaches c through the domain of f, exists and is equal to f(c).

  9. [Proving continuity using sequences] How do we prove that f is continuous?

    For any sequence: if sequence → a, then f(sequence) must → f(a).

  10. [Proving continuity using sequences] How do we prove that f is discontinuous?

    Just find one sequence such that: sequence → a AND f(sequence) ↛ f(a).

  11. [Notation] What does ↛ stand for?

    ↛ stands for "doesn't tend to".

  12. [Notation] Why should we memorize xn → x notation (instead of just using the lim notation)?

    Because it's used extensively not only in OU books, but also on Wikipedia.

  13. [Notation] What does this mean in lim notation?

  14. [Notation] How would you write it in alternative notation?

  15. [Proving continuity using sequences] Name 2 sequences handy when we're proving discontinuity at a.

    (a – 1/n) - for approaching from the left (a + 1/n) - for approaching from the right