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

Question

Aurora Alvarado · Accepted Answer

Yes.

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.

Evelyn Agard · Answer

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

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.

Christopher Agresta · Answer

undefined Yes, a function can be continuous and non-differentiable on a given domain. An example of such a function is the absolute value function $ f(x) = |x| $. This function is continuous everywhere, including at $ x = 0 $, but it is not differentiable at $ x = 0 $ because it has a sharp corner or "kink" at that point. This means that the derivative of the function does not exist at $ x = 0 $.