Uniform convergence of piecewise continuous functions

In summary, the conversation discusses the concept of piecewise functions and their convergence to a function that is not piecewise continuous. The suggestion of using Fourier series as an example is mentioned, but it is unclear if it would work. The idea that a sequence of continuous functions will have a continuous limit regardless of being piecewise is also brought up. The conversation ends with a request for other examples or suggestions.
  • #1
Demon117
165
1
I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous?

I've thought of letting each of these sequences of functions being piecewise in terms of the Fourier series but I am unsure this really works because I haven't figured out what it would converge to.

Any suggestions or other examples?
 
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  • #2
A sequence of continuous functions which converges uniformly has a continuous limit. I do not see how piecewise would change anything. As all sequence functions are joined at the same point, the limit will be, too.
 

FAQ: Uniform convergence of piecewise continuous functions

What is uniform convergence of piecewise continuous functions?

Uniform convergence of piecewise continuous functions is a mathematical concept that describes the behavior of a sequence of functions. It means that as the number of terms in the sequence increases, the functions become closer and closer to a single continuous function. This is different from pointwise convergence, where each individual point in the functions approaches the limit function.

What is the significance of uniform convergence of piecewise continuous functions?

Uniform convergence of piecewise continuous functions is important because it allows us to approximate complicated functions by simpler ones. This can be useful in many applications, such as numerical analysis and signal processing.

How is uniform convergence of piecewise continuous functions different from uniform continuity?

Uniform continuity refers to the behavior of a single function, while uniform convergence of piecewise continuous functions refers to the behavior of a sequence of functions. In uniform continuity, we are concerned with how close the function is to itself for different inputs, while in uniform convergence, we are concerned with how close the functions are to each other as the number of terms in the sequence increases.

How can we prove uniform convergence of piecewise continuous functions?

To prove uniform convergence of piecewise continuous functions, we can use the Weierstrass M-test. This test states that if we can find a sequence of numbers that bounds the absolute value of each function in the sequence, and this sequence converges, then the original sequence of functions converges uniformly. This is a powerful tool for proving uniform convergence.

What are some real-world applications of uniform convergence of piecewise continuous functions?

Uniform convergence of piecewise continuous functions has many real-world applications, such as in numerical methods for solving differential equations, approximation of functions in engineering and physics, and in signal processing for filtering and compression. It also has applications in computer graphics, where it is used to approximate complex shapes with simpler ones.

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