What is the difference between differentiability and continuity of a function?

Question

Christopher Agresta · Accepted Answer

See explanation below

A function #f(x)# is continuous in the point #x_0# if the limit: #lim_(x->x_0) f(x)# exists and is finite and equals the value of the function: #f(x_0) = lim_(x->x_0) f(x)# A function #f(x)# is differentiable in the point #x_0# if the limit: #f'(x_0) = lim_(x->x_0) (f(x)-f(x_0))/(x-x_0)# exists and is finite. A differentiable function is always continuous. We can prove it by writing #f(x)# as: #f(x) = f(x_0) + (f(x) - f(x_0))/(x-x_0)(x-x_0)# Passing to the limit for #x->x_0#: #lim_(x->x_0) f(x) = f(x_0) + lim_(x->x_0) ((f(x) - f(x_0))/(x-x_0))*lim_(x->x_0) (x-x_0)# #lim_(x->x_0) f(x) = f(x_0) + f'(x_0)*0= f(x_0)# A function can be continuous but not differentiable, for example: #y = abs(x)# is continuous but not differentiable in #x=0#.

Aurora Alvarado · Answer

undefined Differentiability refers to the existence of the derivative of a function at a given point. A function is said to be differentiable at a point if the derivative exists at that point. Continuity, on the other hand, refers to the smoothness or uninterrupted behavior of a function over its entire domain. A function is continuous at a point if the limit of the function exists at that point and is equal to the value of the function at that point. While every differentiable function is continuous, not every continuous function is differentiable.