Is the Limit Definition of Continuity Equivalent to the Standard Definition?

[Freye]

Hey guys,

Continuity is generally expressed as lim x->a f(x)=f(a).
But is it also correct to express it as: lim h->0 f(x+h) - f(x) = 0?
Because that would imply that all numbers around f(x) would have to be very close to f(x), and that is basically what continuity is, no?

 

[gb7nash]

Hey guys,

Continuity is generally expressed as lim x->a f(x)=f(a).
But is it also correct to express it as: lim h->0 f(x+h) - f(x) = 0?
Because that would imply that all numbers around f(x) would have to be very close to f(x), and that is basically what continuity is, no?

Yes, this is fine (assuming f(x) exists)

 

[Freye]

Ok thank you, it really helps me out on a problem I am working on.