Question

What happens at f(c) if

f(c) exists
f'(c) exists
f''(c) does not exist

Answer 1

f(c) exists you may not know whether the function is continuos at c or not...it is just that the value is defined at c

Answer 2

f'(c) exists means that f(x) is continues at x=c

Answer 3

that's incorrect, and this question has nothing to do with continuity. also, i'm asking for what happens when all 3 conditions are satisfied.

Answer 4

What I have just wrote is correct...but may be it is not related to your question

Answer 5

knowing f(c), f'(c) and f''(c) has no bearing on a function's continuity. even if f'(c) were DNE, the function could still be continuous.

Answer 6

f'(c) exist implies continuity at x=c.
since f''(c) does not exist, it can mean that concavity of the function f does not change at x=c.