Differentiability/Continuity Question

[ZachDodsonCali]

Can someone plz help me with this question? I think II is correct because the equation has an answer, 5, meaning it’s differentiable at x=2. And I think I remember that differentiability implies continuity, so I think the answer is C. I’m not sure about III though.

 

[Steven G]

View attachment 13589
Can someone plz help me with this question? I think II is correct because the equation has an answer, 5, meaning it’s differentiable at x=2. And I think I remember that differentiability implies continuity, so I think the answer is C. I’m not sure about III though.

I think I remember that differentiability implies continuity is correct. So I is correct
Based on the choices it is clear that III can't be true as there is no choice that includes I, II and III.

Now try to think of an example where f(x) is differential at x=2, but the derivative of is NOT continuous at x=2.