why the following sentence is false?

Question

False:A function f is differentiable at c if its graph has a
tangent line at the point (c; f(c)).

on the other hand
True: If the graph of the function f has a tangent line at the point (c,f(c)) then f is differentiable at c.

Answer 1

If there is a tangent line at (c,f(c)) then the function is differentiable at (c,f(c)) as stated by the second statement.

However, if there is another point along the function f(c) that has no tangent point, the function is not differentaible at that point. The original statement is not neccessarily false, but if there is any point that can't be differentiated in the function, then it is false.

Answer 2

I think that my Calculus professor said that this function

f(x) = x sin (pi/x) if x !=0 and f(0) = 0

is differenciable in x=0 but there is not a tangent line in x=0.

Let's prove that he was right

lim [f(x) - f(0)]/(x-0) = f'(0) = 0
x ->0

And yes, he was... Just draw the function and you will see that there is not a tangent line in x = 0

Ilusion

Answer 3

I have the feeling that there's a typo somewhere in this ☺

Doug