Question

Can a function be continuous and non-differentiable on a given domain??

Answer 1

Yes.

Explanation:

One of the most striking examples of this is the Weierstrass function, discovered by Karl Weierstrass which he defined in his original paper as:

`sum_(n=0)^oo a^n cos(b^n pi x)`

where `0 < a < 1`, `b` is a positive odd integer and `ab > (3pi+2)/2`

This is a very spiky function that is continuous everywhere on the Real line, but differentiable nowhere.

Answer 2

Yes, if it has a "bent" point. One example is `f(x)=|x|` at `x_0=0`

Explanation:

Continuous function practically means drawing it without taking your pencil off the paper. Mathematically, it means that for any `x_0` the values of `f(x_0)` as they are approached with infinitely small `dx` from left and right must be equal:

`lim_(x->x_0^-)(f(x))=lim_(x->x_0^+)(f(x))`

where the minus sign means approaching from left and plus sign means approaching from right.

Differentiable function practically means a function that steadily changes its slope (NOT at a constant rate). Therefore, a function that is non-differentiable at a given point practically means that it abruptly changes it's slope from the left of that point to the right.

Let's see 2 functions.

`f(x)=x^2` at `x_0=2`

Graph

graph{x^2 [-10, 10, -5.21, 5.21]}

Graph (zoomed)

graph{x^2 [0.282, 3.7, 3.073, 4.783]}

Since at `x_0=2` the graph can be formed without taking the pencil off the paper, the function is continuous at that point. Since it is not bent at that point, it's also differentiable.

`g(x)=|x|` at `x_0=0`

Graph

graph{absx [-10, 10, -5.21, 5.21]}

At `x_0=0` the function is continuous as it can be drawn without taking the pencil off the paper. However, since it bents at that point, the function is not differentiable.