Approximation of continuous functions by differentiable ones

In summary, the problem presents a function g(x) that is defined by integrating a continuous function f over a certain interval. The task is to show that g(x) is continuously differentiable and that if f is uniformly continuous, then there exists a positive value delta, delta1, such that the supremum of the absolute value of f(x) minus g(x) is less than epsilon for all delta less than or equal to delta1.
  • #1
cooljosh2k2
69
0

Homework Statement


Let f: R-->R be continuous. For δ>0, define g: R-->R by:

g(x) = (1/2δ) ∫ (from x-δ to x+δ) f

Show:

a) g is continuously differentiable

b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1

The Attempt at a Solution



Please help me, I am confused
 
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  • #2
cooljosh2k2 said:

Homework Statement


Let f: R-->R be continuous. For δ>0, define g: R-->R by:

g(x) = (1/2δ) ∫ (from x-δ to x+δ) f
Frankly, I don't understand this part. If f is not linear, or at least piecewise-linear, g will depend upon both x and [itex]\delta[/itex]. Was there a "[itex]\lim_{\delta\to 0}[/itex]" in the definition?

Show:

a) g is continuously differentiable

b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1


The Attempt at a Solution



For part a, according to the fundamental theorem of calculus, if i integrate the given integral, i get a value of 1. How do i show that g is continuously differentiable though.
?? How do you arrive at the conclusion that the integral is identically 1? The value of the integral will strongly depend upon f.

and for part b, I am just trying to figure out part a before tackling part b.
 
  • #3
No, there was no limit defined in the problem, i typed it exactly how my prof. worded it.

And you are right about my conclusion for the integral, had a brain cramp and figured that f is equal to 1, which would make my integrating very easy. So completely disregard that i said that.
 

Related to Approximation of continuous functions by differentiable ones

1. What is the purpose of approximating continuous functions by differentiable ones?

The main purpose of approximating continuous functions by differentiable ones is to simplify the analysis of the function. Continuous functions are often difficult to work with mathematically, but by approximating them with differentiable functions, we can use the tools of calculus to better understand the behavior of the function.

2. How is a continuous function approximated by a differentiable one?

Continuous functions can be approximated by differentiable ones using a variety of methods, such as Taylor series, Fourier series, and piecewise polynomial interpolation. These methods involve finding a series of simpler functions that converge to the original continuous function.

3. Are there any limitations to approximating continuous functions by differentiable ones?

Yes, there are some limitations to this process. One limitation is that the accuracy of the approximation depends on the smoothness of the original function. If the function is not very smooth, the approximation may not be accurate. Another limitation is that the differentiable function may not capture all the features of the original function, such as sharp corners or discontinuities.

4. How do I know if my approximation is good enough?

The accuracy of an approximation can be evaluated by calculating the error between the original function and the approximating function. This can be done by using various error measures, such as the maximum error or the root mean square error. A smaller error indicates a better approximation.

5. In what real-world applications is approximating continuous functions by differentiable ones useful?

Approximating continuous functions by differentiable ones is useful in many fields of science and engineering. This includes applications in physics, economics, signal processing, and computer graphics. For example, in physics, the motion of a particle can be approximated by a differentiable function to predict its future positions. In computer graphics, smooth curves and surfaces can be approximated by differentiable functions to create realistic images.

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