Diferencovatelná

Diferencovatelná is a term used in mathematics, particularly in calculus, to describe a function that can be differentiated. A function is differentiable if its derivative exists at a given point or over an interval. The derivative of a function at a point represents the instantaneous rate of change of the function at that point, which geometrically corresponds to the slope of the tangent line to the function's graph at that point.

For a real-valued function of a single real variable, $f(x)$, to be differentiable at a point $x_0$, the limit

$$ \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} $$

must exist. If this limit exists, it is denoted by $f'(x_0)$. The existence of this limit implies that the graph of the function is locally linear around the point $x_0$. In simpler terms, the function has a well-defined tangent line at that point, and it does not have any sharp corners, cusps, or vertical tangents.

If a function is differentiable at every point in an open interval, it is said to be differentiable on that interval. Differentiability is a stronger condition than continuity. While all differentiable functions are continuous, not all continuous functions are differentiable. For example, the absolute value function, $f(x) = |x|$, is continuous everywhere but is not differentiable at $x=0$ due to the sharp corner in its graph.

In higher dimensions, the concept extends to functions of multiple variables, where differentiability is related to the existence of partial derivatives and the ability to approximate the function locally by a linear transformation (the Jacobian). A function that is differentiable at a point in this multivariate context is also continuous at that point.