Can someone please explain to me how the following statement is true? This is fr
ID: 1941812 • Letter: C
Question
Can someone please explain to me how the following statement is true? This is from the wikipedia page on discontinuities. They have the following picture,
and under this picture is the statement, "The derivative of this curve has a jump discontinuity." I thought that Darboux's theorem tells us that if a function is the derivative of a differentiable function then it has no jump discontinuties. What am I missing or should the statement say that it has an essential discontinuity? WIll rate lifesaver to anyone who can either correct my thinking or ensure me that the statement is false. Please explain with example if I am wrong.
Explanation / Answer
the darboux theorem which u are stating is just a part of it ...... There is an extension to it ...... But even when f' is not continuous, Darboux's theorem places a severe restriction on what it can be. Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the left hand and right hand limits does not exist. An example of a Darboux function that is discontinuous at one point, is the function .