Question

I study mathematics-economics in the second year of a bachelor programme: I’m one month into the course “Analysis I”. I would like to get more familiar with certain subjects. For a reasonable answer for this Q&A, I’d like a formal definition, a simple example/proof and summary of the mentioned subject.

The subject is differentiability of one real function of one variable.

Answer 1

Definition:-

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

More generally, if x₀ is a point in the domain of a function f, then f is said to be differentiable at x₀ if the derivative f ′(x₀) exists. This means that the graph of f has a non-vertical tangent line at the point (x₀, f(x₀)). The function f may also be called locally linear at x₀, as it can be well approximated by a linear function near this point.

For a function of one variable, the derivative gives us the slope of the tangent line, and a function of one variable is differentiable if the derivative exists. For a function of two variables, the function is differentiable at a point if it has a tangent plane at that point. But existence of the first partial derivatives is not quite enough, unlike the one-variable case.

Theorem:-

If f(x,y) has continuous partial derivatives fx(x,y) and fy(x,y) on a disk D whose interior contains (a,b), then f(x,y) is differentiable at (a,b).

Theorem:-

If f is differentiable at (a,b), then f is continuous at (a,b).