If a function is differentiable in a given interval or at a point?

Question

should it necessarily be also continuous in that interval or point?

Answer 1

yes. A fundamental rule in calculus is differentiability constitutes continuity. Anytime you can differentiate a function, it always is continuous. But a continous function is not always differentiable.

Answer 2

y= f(x) as f´(x) exists lim delta y/delta x as (D = delta)
Dx=>0 So Dy must => to zero because if not the limit would be infinite or not existant..So your diffentiable function is continuos

Answer 3

Yes. "Should" is an understatement.