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Give me a function that is piecewise continuous but not piecewise smooth
jmm said:|x| is defined piecewise as abs(x)={-x,x<0; x,x>0} which is continuous but not smooth at x=0. And is perhaps a little bit easier to visualize than a fractal :D
jmm said:|x| is defined piecewise as abs(x)={-x,x<0; x,x>0} which is continuous but not smooth at x=0. And is perhaps a little bit easier to visualize than a fractal :D
Piecewise continuity means that a function is continuous over intervals, but may have discontinuities at certain points where the intervals meet. On the other hand, piecewise smoothness means that a function has continuous derivatives over each interval. Therefore, a function that is piecewise continuous but not piecewise smooth will have discontinuities at certain points, but its derivatives will still be defined and continuous over each interval.
One example is the absolute value function, f(x) = |x|, which is continuous over the intervals [-∞, 0) and (0, ∞), but has a sharp point at x=0 where the intervals meet. The derivative of this function is not defined at x=0, making it piecewise continuous but not piecewise smooth.
To prove that a function is piecewise continuous but not piecewise smooth, we can use the definition of continuity and piecewise functions. We need to show that the function is continuous over each interval, but has a discontinuity at the points where the intervals meet. We also need to show that the derivatives of the function are not continuous at these points.
Yes, there are many real-world applications of such functions. One example is the velocity of a moving object, which can be modeled by a piecewise continuous function. The object may experience sudden changes in velocity at certain points, making it piecewise continuous but not piecewise smooth. This can be observed in situations such as a car accelerating and suddenly braking or a rollercoaster going up and down.
To make a piecewise continuous function also piecewise smooth, we need to ensure that the derivatives of the function are continuous at the points where the intervals meet. This can be achieved by carefully choosing the function and its intervals, and making sure that the derivatives match at the points where the intervals meet. In some cases, we may need to use a different piecewise function that better fits the given data to make it piecewise smooth.