Definition of Differentiable:
Differentiability is in context to a function, we will always be considering a function and then will be telling about its differentiability.
So let's see the definition of differentiable
“Let f(x) be a real valued function defined on (a,b) and let c∈(a,b) then f(x) is said to be derivable or differentiable function at x = c, iff following exists finitly:”
$$ \lim\limits_{x \to c} \frac{f(x)-f(c)}{x-c} $$
We have a function given to us and that is a real valued function f(x)and it is defined on the open interval (a,b). Now, there exists a point "c" on that interval of (a,b) i.e. c∈(a,b), then the function will be differentiable if it exists on the above limit for finite quantity.
Notation of Differentiable Function:
Now how to represent differentiability if I have a function f(x) or f(c) then its notation will be:
$$ f'(x) = f'(c) = Df(c) = \left| \frac{df(x)}{dx} \right|_{x=c} $$
Condition for Differentiability:
We talk earlier that if limit exists, what limit exists means the left hand limit should be equal to the right hand limit and in case of differentiability we say that the left hand derivative should be equal to the right hand derivative then we say that the limit exists finitely means it should be a finite quantity. So, the main idea is to find the derivative of a function.
Mathematically,
$$ \lim\limits_{x \to c^-} \frac{f(x)-f(c)}{x-c} = \lim\limits_{x \to c^+} \frac{f(x)-f(c)}{x-c} $$ $$ \lim\limits_{h \to 0} \frac{f(c-h)-f(c)}{c-h-c} = \lim\limits_{h \to 0} \frac{f(c+h)-f(c)}{c+h-c} $$ $$ \lim\limits_{h \to 0} \frac{f(c-h)-f(c)}{-h} = \lim\limits_{h \to 0} \frac{f(c+h)-f(c)}{h} $$
Let’s try to deep understanding by some practical meaning
Differentiable Function Example:
Consider we have a function:
$$ f(x) = \begin{cases} x, & x \le 1 \\ x^3, & x \gt 1 \end{cases} $$
Now for testing differentiability, Let's find the derivative of a function
$$ f'(x) = \begin{cases} 1, & x \le 1 \\ 3x^2, & x \gt 1 \end{cases} $$
Now, let’s see what is the left hand derivative and also right hand derivative
The left-hand derivative of a function is:
$$ \lim\limits_{x \to 1^-} f'(x)=1 $$
Also, The right-hand derivative of a function is:
$$ \lim\limits_{x \to 1^+} f'(x)=3(1)^2 $$ $$ \lim\limits_{x \to 1^+} f'(x)=3 $$
So, both side limits are not the same which shows that the function is not a differentiable function.