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

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.

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.