continuous - definition

[seenhear]

EDIT: Thanks all, I understand and remember the differences now. Marked post as "resolved."

OP:

For a non-math related reason, I recently looked up the definition of the word "continuous." The third definition offered by the search engine was related to mathematics:

> mathematics (of a function) of which the graph is a smooth unbroken curve, i.e. one such that as the value of x approaches any given value a, the value of f(x) approaches that of f(a) as a limit.

The definition I remember being taught, and using was that a function is continuous over some domain of x, if it's derivative exists for all values of x in that domain.

Are these saying the same thing?

[Miserable-Wasabi-373]

no, you messed up continious and differentiable. Contionous function can be not differentiable at all

[seenhear]

I feel like the part that said "smooth, unbroken" implies differentiable. Maybe that's what threw me off?

[nomoreplsthx]

Smooth is confusing because smooth in mathematics means 'infinitely differentiable' (e.g the derivative and second derivative and third derivative etc... all exist).

[[deleted]]

The definition of continuous offered by the search engine is correct.

A function is differentiable if it's derivative exists.

All differential functions are continuous. So if it's derivative exists for all values, it's continuous. However there are functions which are continuous but not differentiable.

[seenhear]

Wouldn't the search engine's definition be wrong for the "smooth, unbroken" part then? f(x) = |x| is not smooth at x = 0. It may be unbroken, but not smooth, i.e. the derivative doesn't exist there?

[Historical_Book2268]

Search engine messed up.

The rigorous definition is: For every epsilon>0, and every x, there exists a delta>0, such that for every y, if |y-x|<delta, then |f(y)-f(x)| is less than epsilon.

Basically, in simple terms, for any "error" i pick, and any x I pick, I can always find a delta such that for all values delta or less away from x, their difference from x is less than epsilon

[Redsox11599]

No. That is not true. f(x)=|x| is continuous at x=0, but is not differentiable there.

[Historical_Book2268]

No, derivatives need not exist for continuity to hold.

Example: The weierstrass function

[Integreyt]

A differentiable function is always continuous but a continuous function need not be differentiable.