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?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
?? How do you arrive at the conclusion that the integral is identically 1? The value of the integral will strongly depend upon 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
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.
and for part b, I am just trying to figure out part a before tackling part b.
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.
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.
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.
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.
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.