Surely you can do the piecewise continuous part? It's just matching up if the curves are continous. So, for example, sin(1/x) has to have two limits: \(\displaystyle \lim_{x \to 0^+} f(x) = 0\) (since f(0) = 0) and \(\displaystyle \lim_{x \to 0^-} f(x) = 0\).comfortablynumb said:I do not know to start. Here is the problem.Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here $x\neq0$ is in the interval $[-1,1]$ and $f(0)=0$ in all cases.
1. $f(x)=sin(\frac{1}{x})$
2. $f(x)=xsin(\frac{1}{x})$
3. $f(x)={x}^{2}sin(\frac{1}{x})$
4. $f(x)={x}^{3}sin(\frac{1}{x})$ .
Piecewise continuous functions are continuous on each interval within their domain, but may have discontinuities at the endpoints of these intervals. Piecewise smooth functions, on the other hand, are not only continuous on each interval, but also have continuous derivatives on each interval.
To determine if a function is piecewise continuous, you can check for any discontinuities at the endpoints of the intervals within its domain. To determine if a function is piecewise smooth, you can check if it has continuous derivatives on each interval within its domain.
Yes, a function can be both piecewise continuous and piecewise smooth. This means that it is continuous on each interval within its domain and also has continuous derivatives on each interval.
A real-world example of a piecewise continuous function is a piecewise linear function, such as a staircase function. A real-world example of a piecewise smooth function is a piecewise polynomial function, such as a cubic spline used in computer graphics.
Piecewise continuous and piecewise smooth functions are used in science to model real-world phenomena that may have discontinuities or changes in behavior. They are commonly used in physics, engineering, and computer science to represent functions that are not continuous or smooth throughout their entire domain.