Real valued functions A,B,C,D

Introduction to Real Valued Functions

  • Focus on foundational concepts in higher mathematics: limits and continuity.

  • Introduction to General Topology, an advanced mathematics domain beyond this course.

Generalities

Definition of a Real-Valued Function

  • A function is defined as ( f: D \to R)

    • Where ( D ) is a non-empty subset of ( R ):

    • Notation: ( x \mapsto f(x) = y )

Algebraic Operations on Functions

  • Given functions ( f: D \to R ) and ( g: D \to R ):

    1. Sum of Functions:

      • ( (f + g): D \to R)

      • Defined as ( (f + g)(x) = f(x) + g(x)), for all ( x \in D ).

    2. Product of Functions:

      • ( (f.g): D \to R )

      • Defined as ( (f.g)(x) = f(x) . g(x) ), for all ( x \in D ).

    3. Quotient of Functions:

      • Assuming ( g(x)
        eq 0 ):

      • Defined as ( \frac{f}{g}: D \to R )

      • Defined as ( \frac{f}{g}(x) = \frac{f(x)}{g(x)}), for all ( x \in D ).

Types of Functions

Even and Odd Functions

  • A function ( f: D \to R ) is:

    • Even if ( f(-x) = f(x) ), for all ( x \in D ).

    • Odd if ( f(-x) = -f(x) ), for all ( x \in D ).

Periodic Functions

  • A function ( f: D \to R ) is periodic if:

    • There exists ( T > 0 ) such that ( f(x + T) = f(x) ) for all ( x \in D ).

    • Fundamental period is the smallest period ( \tau > 0 ).

    • Examples:

      • ( f(x) = \cos x ), ( g(x) = \sin x ) have ( \tau = 2\pi ).

      • ( h(x) = \tan x ) has period ( \tau = \pi ).

Intervals and Neighborhoods

  1. For ( x_0 , \in R ): Centered open neighborhood is ( ]x_0 - r, x_0 + r[ ), with ( r > 0 ).

  2. For ( x_0 = +\infty ): Open neighborhood is ( ]A, +\infty[ ).

  3. For ( x_0 = -\infty ): Open neighborhood is ( ]-\infty, B[ ).

  • Remark: ( x \in ]x_0 - r, x_0 + r[ \Leftrightarrow |x - x_0| < r ).

Limits

Definition of Limit

  • For ( x_0 , \in R ), limit ( l , \in R ):

  • A function ( f: D \to R ) tends toward ( l ) as ( x ) approaches ( x_0 ) if:

    • ( \forall \varepsilon > 0, \exists \delta > 0) such that ( 0 < |x - x_0| < \delta \Rightarrow |f(x) - l| < \varepsilon ).

  • Alternatively, using neighborhoods:

    • For each centered neighborhood ( U(l)), there exists a centered neighborhood ( V(x_0) ) such that ( f(V(x_0) - {x_0}) \subset U(l)).

Theorem of Unicity

  • If ( f ) tends toward two different limits ( l_0 ) and ( l'' ) as ( x ) approaches ( x_0 ), then ( l_0 = l'' ).

Notation of Limit

  • Limit notation: ( \lim_{x \to x_0} f(x) = l ) or ( \lim_{x \to x_0} f = l ).

Limits at Infinity

  • ( \lim_{x \to +\infty} f(x) = +\infty ) if for every ( A > 0 ) there exists ( \delta > 0 ) such that ( 0 < |x - x_0| < \delta \Rightarrow f(x) > A ).

Remark on Neighborhoods

  • General limit definition: for ( x_0, l \in R ), we say ( f(x) \to l ) when ( x ) approaches ( x_0 ) if:

    • For each neighborhood ( U(l) ), there exists neighborhood ( V(x_0) ) such that ( f(V(x_0) - {x_0}) \subset U(l) .

Theorem on Algebraic Operations on Limits

  • If ( \lim_{x \to x_0} f = l_0 ) and ( \lim_{x \to x_0} g = l'' ):

    1. ( \lim_{x \to x_0} (f + g) = l_0 + l'' )

    2. ( \lim_{x \to x_0} (f.g) = l_0 . l'' )

    3. ( \lim_{x \to x_0} \frac{f}{g} = \frac{l_0}{l''} ), if ( g ) is not 0.

  • Indeterminate Forms:

    • Operations such as ( \frac{0}{0} ), ( \infty - \infty ), etc. require further investigation.

Theorem on Limits of Functions and Real Sequences

  • The limit of a function at a point is equivalent to the limit of the function at the limit of any sequence converging to that point.

One-Sided Limits

  • Definition of right limit:

    • If for some interval ( ]x_0, x_0 + r[ ) with ( r > 0 ) exists ( l \in R ) such that ( \forall \varepsilon > 0 ), there exists ( \delta > 0 ) satisfying the limit condition.

  • Definition of left limit:

    • Similar approach for an interval ( ]x_0 - r, x_0[ ).

  • Theorem: The existence of a limit point ensures that both one-sided limits exist and are equal.

Remarks on Computation of Limits

  1. If ( \lim_{x \to x_0} f = l \in R ) and ( g(x) ) is bounded, then ( \lim_{x \to x_0} f.g = 0.
    2. If ( f(x) \leq h(x) \leq g(x) ) near ( x_0 ) and both ( f ) and ( g ) converge to the same limit, then ( \lim_{x \to x_0} h = l ).

Continuity and Continuous Functions

Definition of Continuity

  • A function ( f: D \to R ) is continuous at ( x_0 ) if:

    • There is a centered neighborhood ( V(x_0) ) such that:

    • ( \forall V(f(x_0)), \exists U(x_0) \subset D ) where ( f(U(x_0)) \subset V(f(x_0)).

    • This implies ( \lim_{x \to x_0} f(x) = f(x_0) ).

Definition of Continuity on a Domain

  • A function is continuous over ( D ) if it is continuous at every point of ( D ).

One-Sided Continuity

  • A function is continuous at ( x_0 ) on the right if:

    • There exists a neighborhood such that ( f([x_0, x_0 + r[) \subset V(f(x_0)) ).

  • A function is continuous at ( x_0 ) on the left if:

    • There exists a neighborhood such that ( f(]x_0 - r, x_0]) \subset V(f(x_0)) ).

Theorems

  1. A function is continuous at a point if it is continuous from both left and right.

  2. The property of continuity is crucial to study the behavior of functions.

Examples of Continuity

  • Study the continuity of various functions at specified points and intervals to see practical applications of the definitions and theorems.