What is differentiability and continuity?

Hint: Now a function is said to be continuous if the graph of the function does not break. Hence the graph of the function is continuous. Similarly differentiability of a function tells us about existence of derivative of the function at all points on the function.

Complete step by step answer:
Now let us first understand the concept of continuity.
A function is continuous at a point if the function does not break at the point.
Algebraically we say that if \[\underset{x\to c}{\mathop{\lim }}\,f\left( x \right)=f\left( c \right)\] .
Now the definition tells us that the value of the function f as x tends to c is $f\left( c \right)$ .
Hence the function does not break at x = c. Hence the function is continuous at x = c.
If the function is continuous at all points in domain then the function is said to be a continuous function.
Example of continuous functions are $\sin x,{{x}^{2}},\dfrac{1}{x}$
Now let us understand the concept of differentiability.
To understand this we will first define derivatives of a function.
Derivative of a function at point x is given by $\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$ and is denoted by $f'\left( x \right)$ .
Now the derivative of function at point x gives the slope of the curve at point x.
Now a function is called to be differentiable if derivative of function exists at all points in the domain.
Some examples of Differentiable functions are $\sin x,{{x}^{n}},\ln x$ .

Note:
Now note that if a function is differentiable this means the function is continuous as a function needs to be continuous first to be Differentiable. But if the function is continuous this does not mean that the function is differentiable. An example of the function which is continuous but not differentiable is $\left| x \right|$ as the function is not differentiable at $x=0$ .