A list of questions that came to mind dealing with differentiation. They all came at the same time, so I hope the community will accept me listing them as one question. If not, I will separate them.
1) What does the derivative of a piecewise smooth function look like? Specifically at the end points of an interval, where there is a jump discontinuity, the function does not have a derivative because the secant lines from both sides don't approach the same tangent line. So does that mean there is a removable discontinuity in the derivative function at these points?
2) Does a removable discontinuity (with no black dot, if you will, above or below it) mean that the function is undefined there or indeterminate? What's the difference between 7/0 and 0/0?
3) How is it possible for a derivative function to have a jump discontinuity where the one sided limit at the discontinuity of the derivative function equals the function value at that point, and the other one sided limit doesn't equal the function value (an open and solid dot graphically)? The derivative is the limiting value of the secant lines. If the derivative has a value at one point, the secant lines from both sides of the original function approach a common tangent line. Yet in the jump discontinuity, below the solid black dot (if you will), there is an open dot, which says that the limiting value of the secant lines doesn't exist?
4) Is this statement valid: local linearity $\Leftrightarrow$ differentiability? I know that differentiability $\implies$ local linearity (i.e. smoothness).
Best Answer
1) It depends on how the pieces are connected. If it is a jump discontinuity, with both sides approaching the same slope of the tangent line, then the derivative has a removable discontinuity. If the two sides does not go toward each other "parallelly", then the derivative also has a jump. If the two pieces connect with each other, but not very smoothly, the derivative can still have a jump. If they connect with each other with the same slope of the tangent lines, then the derivative is continuous.
2) A removable discontinuity usually happens this way $f(x)=\frac{x^2-1}{x-1}$ where the $x-1$ can be cancelled, or harder to see, $f(x)=\frac{\sin{x}}{x}$, whose limit at $x=0$ is 1, but it is undefined at that point. If you write the constant $7/0$ or $0/0$, they are both undefined. But if they comes from limits, the $7/0$ approaches infinity, the $0/0$ can usually be reduced, for example, by L'Hospital's rule, or other methods.
3) I don't think this could happen, unless the derivative at that side is only defined as the left (or right) derivative, which is not the exact definition.
4) What do you mean by "local linearity"? If it means "locally it is linear", then I guess yes. Though I am not sure if it can be said this way.