The discussion revolves around the continuity and differentiability of a piecewise function defined as f(x)={(2x-1)/|2x-1| for x ≠ 1/2, 0 for x = 1/2. Participants are examining the behavior of the function at the point x = 1/2, particularly focusing on whether the function is continuous and differentiable at that point.
The discussion is active, with participants sharing their interpretations of the graph and questioning the continuity and differentiability of the function at the specified point. Some participants have provided guidance on how to approach the problem, while others are clarifying concepts related to differentiability and continuity.
Participants are navigating through the definitions of continuity and differentiability, with some confusion about the terminology used to describe these concepts. There is an emphasis on ensuring accurate understanding of the relationship between the two properties.
betsinda said:the line from the left approaches (1/2) at +1 and restarts at (1/2) at -1. There is a single point at (0,1/2)
No. It doesn't make any sense to talk about a point or an x value being differentiable. You can say, though, that a function is continuous or differentiable at a point or at some x value.betsinda said:would it then be correct to say that x=1/2 is not differentiable as it is not continuous at that point?
That is correct, so you are not misunderstanding the concept. I have made a couple of edits to what you wrote:betsinda said:based on the fact that well every function is not differentiable, very function that is differentiable is continous. Or am I misunderstanding the concept?
While not every function is differentiable, every function that is differentiable is continous.