Differenciálhatónak

Differenciálhatónak is a term used in mathematics, specifically in calculus, to describe a function that has a derivative at a particular point. A function is considered differentiable at a point if its graph is smooth and does not have any sharp corners, cusps, or vertical tangents at that point.

Formally, a function f(x) is differentiable at a point 'a' if the limit of the difference quotient [f(x) - f(a)] / (x - a) exists as x approaches 'a'. This limit is the derivative of the function at 'a', denoted as f'(a). The derivative represents the instantaneous rate of change of the function at that point, or the slope of the tangent line to the function's graph at that point.

If a function is differentiable at a point, it is also continuous at that point. However, the converse is not always true; a function can be continuous at a point but not differentiable there. A classic example is the absolute value function, f(x) = |x|, which is continuous at x=0 but not differentiable at x=0 due to the sharp corner at that point.

The concept of differentiability is fundamental in calculus and has wide-ranging applications in physics, engineering, economics, and many other fields where understanding rates of change and optimization is crucial. Functions that are differentiable over an entire interval are said to be differentiable on that interval.