Comment 3: Counterexample

Hi Luke,
Today we had a following poll question with the last answer deemed to be correct.
The correct answer should be B rather than D.
Indeed, here is a counterexample of a function that is discontinuous at
x=1
yet nevertheless
◼
  • has the left and right derivatives the same at
    x=1
    , i.e., the function is differentiable at this point.
    • ◼
  • its right and left limits are NOT equal to each other.
    • In[]:=
    f[x_]:=Piecewise[{{
    2
    x
    ,x<=1},{
    2
    x
    +2,x>1}}]
    In[]:=
    {Limit[f[x],x->1,Direction->"FromBelow"],​​Limit[f[x],x->1,Direction->"FromAbove"]}
    Out[]=
    {1,3}
    In[]:=
    Plot[{Callout[f[x],"f[X]",Above],Callout[f'[x],"f'[X]",Above]},{x,0,2},PlotStyle->Thick,AxesLabel->Automatic]
    Out[]=