Question

please tell me what are the conditions of non-differentiability even the functions are continuous?

Answer 1

The condition is that the condition for differentiability is not met. Namely, a function f is differentiable at x = a if the limit as x --> a of
\[ \frac{ f(x) - f(a)}{x-a} \]
exists. (or the limit as h --> 0 ((f(a+h) - f(a))/h) exists)

Hence a function is not differentiable if that limit doesn't exist.

Answer 2

in other words which functions are continuous but not differentiable?

Answer 3

For example, that limit does not exist for the function f(x) = |x| at x = 0.

Answer 4

what is the case of this function is it continuous and differentiable at x=a?

Answer 5

What do you think? It looks pretty obvious.

Hint: if a function is diffble at x = a then it is continuous at x = a. Therefore if a function is not cts at x = a, it is not diffble there.

Answer 6

Hi Rosy,

A function is not differentiable when the "limits" of the function are not defined or continuous. I will try and explain this better.

The part of the function to the left you have drawn above has a point which is undefined (open circle), where as the line on the right has a point which is defined (filled in circle).

The other way to tell if a function is differentiable or not is whether the function is "smooth". As you can see, your two lines of the function do not join at the same 'y' value. Of course, there is an exception to this rule, but I'm sure you will come across this as you continue studying :)

Answer 7

I will try and also provide the explanation of the formula:

So, a function f is continuous at x = a if:
\[\lim_{x \rightarrow a} f(x) = f(a)\]

So, if we were to draw a graph of this function it would look like:



From the function (and hopefully the graph) you will see that the function 'f(x)' at the point 'x' (which can be anywhere on the graph) of the graph corresponds to the f(a) and 'a' value respectively. This suggests that the line is smooth with no gaps or unknown points and is therefore....you guessed it - CONTINUOUS :)