Some doubts in derivative questions containing function

Question

I am learning questions in which we have to prove that a function is differentiable and continuous at a point.

Let I have to find that a function f is differentiable at x = 3.

Then i saw on some place they simply find $\frac{f(x)-f(3)}{x-3}$ at limit x $\to$ 3 and then LHD and RHD.

On some places they find only LHD and RHD.

Similarly for checking function is continuous.

So i want to ask which method is correct?

Answer 1

For checking continuity of a function,

at say c,$f(c) = \lim_{x \to c} f(x)$

For checking differentiability,

You will first have to check whether the function is continuous at that point or not. If function at that point is not continuous, then function will not be differentiable.

And when continuous, you will have to check L.H.D = R.H.D.