continuity, discontinuity, singularity, differentiability

[PG1995]

Hi

Can we say that, in general terms, singularity and discontinuity are the same things? Please help me with it. Thanks.


Helpful Links:
1: https://en.wikipedia.org/wiki/Mathematical_singularity
2: https://en.wikipedia.org/wiki/Differentiable_function



Regards
PG

 

[steveB]

Hi

Can we say that, in general terms, singularity and discontinuity are the same things?

From Wolfram

"A singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. "

"A discontinuity is point at which a mathematical object is discontinuous."

So, it seems the definitions are different, which leads me to say that "in general terms" they "are not the same things".

 

[MrAl]

Hello there,


If you look at the unit step function, at t=0 there is a discontinuity because it jumps from 0 to 1 in zero time. But it's bounded.
On the other hand, if you look at the derivative of the unit step function it goes to infinity at t=0 so it's a singularity. It's not bounded.

 

[steveB]

It's important to point out that the question was asked with the qualifier "in general terms". If we were to talk in "specific terms" we could find examples where the dividing lines are not as clear cut. In real analysis singularities are sometimes classified as discontinuities, and there is also the fact that humans use language in ways that sometimes blur the line even more. For example, someone might say that the function y=1/x is a discontinuous function, when the one point that is troublesome (x=0) is a singularity. Certainly, 1/x is not continuous at x=0, but the question of whether it is proper to say that it is discontinuous is one I can't answer with confidence, but I think it is accepted terminology. For example, wikipedia shows the following definition for continuous function, and 1/x would seem to not meet the definition and hence defaults to a discontinuous function according to this definition.

From wikipedia: Continuous Function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function".