The Topologist's Sine Curve

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In summary, Rudin discusses the calculation of the derivative for the given function, which involves using the absolute value to compute the limit as t approaches zero. This is due to the fact that sin(-x) = -sin(x), making the limit non-negative. The concept of absolute convergence is also mentioned, and it is stated that checking for absolute convergence is more obvious in this example. The use of the absolute value shows that the Newton quotient at zero is limited by a number that approaches zero with t.
  • #1
Bachelier
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On Page 106 in baby rudin (diff. chapter) when he tries to calculate the derivative of the fuction


$$f(x) = \begin{cases}
x^2 sin(\frac{1}{x}) & \textrm{ if }x ≠ 0 \\
0 & \textrm{ if }x = 0 \\
\end{cases}$$

rudin used the absolute value in trying to compute the limit as ##t → 0##

##i.e##

##\left|\frac{f(t) - f(0)}{t - 0}\right| = \left|t \ sin(\frac{1}{x})\right| ≤ |t|##

Why the abs. value?
 
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  • #2
I think it has to see with the fact that sin(-x)=-sin(x) . The limit then will be non-negative. Since is negative
in the 4th quadrant.
 
  • #3
Absolute convergence implies convergence as in chapter 3 of the same book. And it turns out in this example checking absolute convergence is more obvious.
 
  • #4
It just shows that the Newton quotient at zero is squeezed by a number that goes to zero with t.
 
  • #5
Thank you all
 

1. What is the Topologist's Sine Curve?

The Topologist's Sine Curve is a mathematical function that was first introduced by the mathematician Karl Menger in 1926. It is a continuous and non-differentiable function that is used to illustrate various concepts in topology.

2. How is the Topologist's Sine Curve different from a regular sine curve?

The Topologist's Sine Curve is different from a regular sine curve in that it is a non-differentiable function, meaning that it does not have a defined slope at every point. This makes it a useful tool for studying concepts such as continuity and connectedness in topology.

3. What are some properties of the Topologist's Sine Curve?

The Topologist's Sine Curve has a few notable properties, including being continuous but not uniformly continuous, being connected but not path-connected, and having a single point of accumulation but not being compact.

4. How is the Topologist's Sine Curve used in topology?

The Topologist's Sine Curve is often used as a counterexample in topology to illustrate concepts such as continuity, connectedness, and compactness. It can also be used to construct other interesting functions and spaces in topology.

5. Can the Topologist's Sine Curve be visualized?

Yes, the Topologist's Sine Curve can be visualized on a graph. It appears as a classic sine curve with a "break" or discontinuity at a certain point, which is where it is non-differentiable. However, it is important to note that the visualization of this curve may vary depending on the scale and precision of the graph.

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