The Glue Function different properties An Example of How - PowerPoint PPT Presentation

The Glue Function Niko Schultz Function and sequence: The Glue Function different properties An Example of How Mathematicians Think Introducing... the glue function! Let’s make our function Niko Schultz smooth September 5, 2014

The Glue Function Niko Schultz Function and sequence: 1 Function and sequence: different properties different properties Introducing... the glue function! 2 Introducing... the glue function! Let’s make our function smooth 3 Let’s make our function smooth

The Glue Function Niko Schultz Problem Function and sequence: different Find a continuous function f : R → R such that properties lim n → + ∞ f ( n ) = ∞ , but lim x → + ∞ f ( x ) does not exist. Introducing... the glue function! Let’s make our function smooth

The Glue Function Niko Schultz Problem Function and sequence: different Find a continuous function f : R → R such that properties lim n → + ∞ f ( n ) = ∞ , but lim x → + ∞ f ( x ) does not exist. Introducing... the glue function! One function that would have such properties would be one Let’s make that... our function smooth takes consistently increasing values — let’s say, for instance, f ( n ) = n — at all natural numbers takes a constant value — let’s say, 1 — at all half values

The Glue Function Niko Schultz One way to do this is to attach all the points with straight lines. Function and sequence: different properties Introducing... the glue function! Let’s make our function smooth

The Glue Function Niko Schultz One way to do this is to attach all the points with straight lines. Function and Solution sequence: different properties Introducing... the glue  if x = n − 1 1 function! 2   (2 n − 2) x − 2 n 2 + 3 n  if n − 1 Let’s make  2 < x < n  our function   smooth ∀ n ∈ N , f ( x ) = if x = n n (2 − 2 n ) x + 2 n 2 − n  if n < x < n − 1   2    if x = n + 1 1  2

The Glue Function Niko Schultz Function and sequence: different properties Introducing... the glue function! Let’s make our function smooth Figure: A zigzag function.

This function is continuous, but it is not smooth. The Glue Function Niko Schultz Function and sequence: different properties Introducing... the glue function! Let’s make our function smooth

This function is continuous, but it is not smooth. The Glue Function Definition (smooth) Niko Schultz Function and A function is smooth if it is infinitely differentiable. sequence: different properties Introducing... the glue function! Let’s make our function smooth

This function is continuous, but it is not smooth. The Glue Function Definition (smooth) Niko Schultz Function and A function is smooth if it is infinitely differentiable. sequence: different properties Definition (glue function) Introducing... the glue A glue function or interpolating function is a function that function! Let’s make glues together piecewise functions smoothly. our function smooth

This function is continuous, but it is not smooth. The Glue Function Definition (smooth) Niko Schultz Function and A function is smooth if it is infinitely differentiable. sequence: different properties Definition (glue function) Introducing... the glue A glue function or interpolating function is a function that function! Let’s make glues together piecewise functions smoothly. our function smooth Theorem There exists a smooth glue function χ 1 such that � 0 if x < − 1 χ 1 ( x ) = 1 if x > 1

The Glue Function Niko Schultz Function and sequence: different properties Introducing... the glue function! Let’s make our function smooth Figure: The glue function χ 1 ( x ).

The Glue Function Problem Niko Schultz Find a smooth function f : R → R such that Function and sequence: different � x 2 if x < 1 properties f ( x ) = sin x if x > 2 Introducing... the glue function! Let’s make our function smooth

The Glue Function Problem Niko Schultz Find a smooth function f : R → R such that Function and sequence: different � x 2 if x < 1 properties f ( x ) = sin x if x > 2 Introducing... the glue function! Let’s make our function Use a one-half scale of the glue function to make this piecewise smooth function smooth.

The Glue Function Problem Niko Schultz Find a smooth function f : R → R such that Function and sequence: different � x 2 if x < 1 properties f ( x ) = sin x if x > 2 Introducing... the glue function! Let’s make our function Use a one-half scale of the glue function to make this piecewise smooth function smooth. Solution f ( x ) = x 2 · (1 − χ 1 2 ( x − 1 . 5)) + sin x · χ 1 2 ( x − 1 . 5)

The Glue Function Niko Schultz Function and sequence: different properties Introducing... the glue function! Let’s make our function smooth

The Glue Function Niko Schultz Function and sequence: different properties Introducing... the glue function! Let’s make our function smooth Figure: A smooth zigzag function.