7 Real functions Flashcards
Definition 7.1:
Graph of a function
Let I ⊆ R and consider a function f: I → R where I⊆R.
The graph of f is the set {(x, f(x)) : x ∈ I} ⊆ R×R = R²
Definition 7.2:
Sequentially continuous function
Given f: I→R, and x₀ ∈ I, f is sequentially continuous in I if for every (xₙ)n∈N where xₙ∈I for all n∈N,
* if lim.n→∞(xₙ) = x₀ ⇒ lim.n→∞[ f(xₙ) ] = f(x₀).
Theorem 7.1:
Imtermediate value theorem
Assume:
* f: [a,b]→R, f is sequentially continuous in [a,b]
* f(a) ≤ y ≤ f(b) or f(a) ≥ y ≥ f(b)
Then:
* there exists c∈[a,b] such that f(c) = y.
Corollary 7.1:
n-th roots
Let n ∈ N and a > 0. Then there exists x > 0 such that xⁿ = a.
Definition 7.3
Exponential function
For x ∈ R define exp: R→R
exp(x) = ∑∞n=0 [ xⁿ/n! ].
Proposition 7.1:
exp(x+y)
For all x,y ∈R,
exp(x+y) = exp(x) exp(y).
Corollary 7.2
exp(x) > ?
exp(x) > 0 for all x ∈ R.
Proposition 7.2:
Is exp sequentially continuous?
The function exp: R → R is sequentially continuous.
Corollary 7.3:
exp() and injectivity
The exponential function is injective and attains every value in (0, ∞).
Definition 7.4:
Natural logarithm
The inverse function of exp : R → (0, ∞) is called the natural logarithm and denoted
by log.
Proposition 7.3:
log(uv)
For all u,v > 0,
log(uv) = log u + log v.
Definition 7.5:
aˣ
For a > 0 and x ∈ R, define aˣ = exp(x log a).
Proposition 7.4:
1. aˣʸ
2. (ab)ˣ
3. aˣ⁺ʸ
- aˣʸ = (aˣ)ʸ for all a > 0 and x,y ∈ R.
- (ab)ˣ =aˣbˣ for all a,b > 0 and x ∈ R.
- aˣ⁺ʸ =aˣaʸ for all a > 0 and x,y ∈ R.
Definition 7.6:
cos(x)
For all x ∈ R
cos(x) = ∑∞n=0[ ((-1)ⁿ/(2n)!) x²ⁿ ]
Definition 7.6:
sin(x)
For all x ∈ R
sin(x) = ∑∞n=0[ ((-1)ⁿ/(2n+1)!) x²ⁿ⁺¹ ]
