Question

"if f is differentiable then f is continuous"
the converse:
"if f is continuous then f is differentiable"
converse is not true. so I am going to proof it with counterexample.
f(x) = 3x+5; x<1 and 10-2x; x>= 1
f is continuous at x = 1
but not differentiable at x = 1.

my question is:
there is f(x) = x^2, this function is continuous and differentiable at x=2,
so how to defense my previous statement which is converse is not true?

Answer 1

The better wording is converse is not ALWAYS true. Th statement and converse both holds for x^2.

Answer 2

It might help you to look up the concept of continuity and search there for examples of functions which are continuous but not differentiable at a certain x value.

The absolute value function is one such function; it is continuous for all x, but not differentiable at x=0. Draw this function and see if you can figure out why that is true.

Answer 3

A good example to take is f(x)=|x|

The function is differentiable at all points except 0 and found to be continuous at all non zero points. But even though it is continuous at 0 it is not differentiable at 0. I hope you got the point.

Answer 4

right now, I am learning logical mathematics, so I have to proof it not with graphic, but argument such as
(p->q) =! (q->p)

Answer 5

Apply Left hand derivative and right hand derivative at x=0 for f(x)=|x| You will get different results. This means derivative doesn't exist even though function is continuous at that point.

Answer 6

The statements become true in the context of \(x=2\).

Answer 7

The same way that \(x^2\geq 0\) is true in the context of \(x\in \mathbb R\).

Answer 8

Yes and that is why I said "The better wording is converse is NOT always true". It could be true at some points whereas it could not hold at other ones.

Answer 9

I think so, your answer is the best. thank you very much

Answer 10

yw :)

Answer 11

thank you for mathmale and wio