Are Functions Continuous at a Cusp or Corner?

[Rozy]

I'm just working through some differentiability questions and have a quick question - are functions continuous at a cusp or corner? I know that functions are not differentiable at cusps or corners because you cannot draw a unique tangent at these points, but I'm not sure about continuity. From my understanding, a function is continuous if at point a if

1. The limit as x approaches a exists
2. f(a) exists or is defined
3. the limit of x approaching a = f(a)

Can you define the function at a cusp or corner and thus have a continuous function?

Thanks!
Rozy

 

[0rthodontist]

Yes, you can.

 

[arildno]

On a side-note, you can construct functions that are everywhere continuous, but nowhere differentiable..

 

[Rozy]

Thanks

Thanks for responding!

 

[HallsofIvy]

For example f(x)= |x| has a "cusp" or corner at x= 0. Of course, f(0)= |0|= 0 so the function is certainly define there- and continuous for all x.