thnx captain and perfecthacker.

ok this is what i understood from what you said perfecthacker. here's my proof from both your help..

you used the sequential definition of continuity? which implies that:

continuity at 0 = limit at 0 exists and equals f(0)?

**PROOF**
sin(1/x) is continuous at

if for any sequence

R, such that

, we have

now let

and

so

if we consider,

we have:

=

=

= 0

but, if we consider,

we have:

=

=

= 1

we see there are points arbitrarily close to 0 where sin(1/x) = 1.

so this means the limit at 0 doesn't exist

as the limit doesnt exist. the function is not continuous. this implies that it is not differentiable at 0. Q.E.D

is this completely correct? and is there anything else i need to add onto this proof? thanks