Question

can anybody tell me how to find continuty n differntiability of a functin...?

Answer 1

If a function is differentiable on an interval, it is also continuous on that interval. A function is basically continuous on every part of the graph, except where there is a vertical asymptote is the tangent line at a point, a "hole" in the graph, or some sort of infinite discontinuity.

For example: the function f(x)=(x^3)/(x-3) is continuous everywhere, except for when x = 3. This is becasue when x = 3, we get 27/0, which is undefined. At least in the terms of a new-linear algebra, but experienced calculus student, that's how continuity is defined on an interval. That interval is important though. You need to be able to figure out whether a function will converge as it approaches certain points, or if it diverges. Maybe it's only defined for complex numbers at a certain point.

If you need things explained in more detail, let me know.

Answer 2

As given the question depends upon where you start from.

The general answer is that you have to recourse to the definitions of continuity and differentiability. So that's the business with limits and whatnot.

Continuity at a certain functional point [ say y = f(x) ] is demonstrated by showing that each ( arbitrarily small ) neighborhood of that point y entirely contains the image of some neighborhood of the point x. Hence you get proofs of the form 'given arbitrary epsilon .... this is the delta for that epsilon'. This is the case for any set type(s), real or complex etc. A function is said to be continuous on some domain set if it is continuous at each point.

Differentiability is not guaranteed by continuity, but continuity is required. Thus a differentiable function must be continuous. Differentiability embodies the idea of slope, or rate of change of one thing with regard to another. So the trick here is that we are looking at the character of a ratio - rise divided by run - in the limit of an arbitrarily small neighborhood. So the proofs tend to have the same flavour as for continuity.

However in practice no one really does any of the above for every function they ever meet. We rely upon the prior work of others who have established theorems like : the sum of two continuous functions is continuous, etc. There are whole classes of functions for which we only need to take on trust certain given behaviours : polynomials, the trig functions, exponentials, and so on.