1.
f (x ):= sin (1 / x ) ,
f ( 0 ):= 0
2.
f (x ):= x · sin (1 / x ) ,
3.
f (x ):= x2 · sin (1 / x ) ,
4.
5.
f (x ):= [x ] · sin ( π · x )
( Continuous, but f is not differentiable at integers x . )
f ( 0 ):= 0
f ( 0 ):= 0
( At x=0 f has a second order discontinuity, f has no one-sided limits at 0. )
( f is a continuous function, but at x=0 f is not differentiable. )
( f is a differentiable function, its derivative, f ' is not continuous at 0 . )
f (x ):= [x ] · sin2 ( π · x )
( Continuous, and differentiable at all x real numbers. )
6.
f (x ):= x3 · ( 2 + sin (1 / x )) ,
f ( 0 ):= 0
7.
f (x ):= x2 · ( 2 + sin (1 / x )) ,
f ( 0 ):= 0
8.
f (x ):= exp (-1 / x2 ] ,
9.
( At x=1/k with integer k f has jump discontinuity
f (x ):= x · [1 / x ] , f ( 0 ):= 1
( that is f has finite left and right limits, but they differ from each other ) . f is continuous at x= 0 . )
f ( 0 ):= 0
( At x=0 f is locally strictly increesing, but there is no r>0, such that
f is increesing in the interval (-r,r) . )
( f is differentiable; at x=0 (glob.min.) the derivative does not change the signe.)
( f is infinite times differentiable at x= 0 (too), and all derivatives hear equal 0 .)