Question

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

Answer 1

Here in this question we have been asked to write the differences between differentiability and continuity of a function. For answering this question we will discuss the definition of differentiability and continuity with examples.

Complete step-by-step solution:
Now considering from the question we have been asked to write the differences between differentiability and continuity of a function.
From the basic concepts we know that a function is said to be continuous over a domain if it is specified for all values in the domain whereas if not specified for some values then it is said to be discontinuous.
The differentiability of a function tells about the existence of the derivative of the function at all points in the given domain.
Examples of differentiable functions are $\cos x,{{x}^{2}}$ .
Mathematically we define continuity as $\displaystyle \lim_{x \to c}f\left( x \right)=f\left( c \right)$ that is the limit of the function $f\left( x \right)$ tends to the value of the function at $c$ when $x$ tends to $c$ .
Examples of continuous functions are $\sin x,{{x}^{2}}$ whereas $\tan x,\dfrac{1}{x}$ are examples of discontinuous functions over real numbers.
Therefore we can say that all the differentiable functions are continuous over the same domain whereas all the continuous functions may or may not be differentiable.

Note: During the process of answering questions of this type we should be sure with the concepts that we are going to apply in between as this is purely a concept based question. Most of the members generally confuse this topic so it's better practice more and more examples from this topic.