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

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Answer 1 · 2017-02-07T10:14:17

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#.