Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. This means that the graph of the function has a smooth, non-vertical tangent line at every point where the function is defined. Such functions are considered smooth because they can be well approximated by a linear function near each point, without any sharp turns, breaks, or cusp.

If a point is inside the area where a function is defined, the function is called differentiable at that point if its derivative exists there. This tells us that the graph of the function has a clear tangent line at that exact point. A function is differentiable over an area if it is differentiable at every point in that area. When a function's derivative is also a continuous function across its domain, it is called continuously differentiable.

For functions with more than one variable, determining differentiability becomes more complex than simply checking for the existence of partial derivatives.

Differentiability of real functions of one variable

A differentiable function is one where we can find a slope at every point along its graph. This means the function doesn't have any sharp corners or breaks. If a function is differentiable, it is also continuous, meaning there are no jumps or gaps in the function's graph.

The article also discusses semi-differentiability, which looks at how we can find slopes at the edges of the function's domain. For more information, see the Semi-differentiability article.

Differentiability and continuity

See also: Continuous function

If a function can be differentiated at a point, it must also be continuous at that point. This means that every differentiable function is continuous everywhere in its domain. However, the opposite is not true — a function can be continuous but not differentiable. For instance, a function with a sharp bend or point might still be continuous but not smooth enough to have a derivative at that spot.

Most everyday functions we use have derivatives almost everywhere. But there are special cases, like the Weierstrass function, which is continuous everywhere but has no derivatives at any point. This shows that while differentiability is common, it is not the norm for all continuous functions.

Differentiability classes

Main article: Smoothness

A function is called continuously differentiable if its derivative exists and is a smooth, unbroken curve itself. Even if a function’s derivative exists everywhere, it might not always be smooth. For example, some functions can have derivatives that jump or behave oddly near certain points.

We also describe functions by how many smooth derivatives they have. A class C¹ function has a smooth first derivative. A class C² function has smooth first and second derivatives. When a function has smooth derivatives of all orders, it is called smooth or class C∞.

Differentiability in higher dimensions

See also: Multivariable calculus and Smoothness § Multivariate differentiability classes

A function with several inputs and outputs is differentiable at a point if it can be well approximated by a straight-line rule near that point. This means the function doesn't have any sharp corners or breaks at that spot.

Even if all the small-direction slopes (partial and directional derivatives) exist at a point, the function might still not be differentiable there. For example, some special functions have all these slopes at a point but still behave in a way that isn't smooth enough to be differentiable.

Differentiability in complex analysis

Main article: Holomorphic function

In complex analysis, a function can be differentiable using rules similar to those for real numbers. This is possible because complex numbers can be divided. When a function is differentiable at a point in the complex plane, it means the function changes smoothly around that point.

Complex-differentiability is a stricter rule than for real functions. If a function is complex-differentiable at a point, it will also be differentiable when viewed as a function of two real variables. However, some functions can be differentiable as two-variable real functions but not as complex functions. These functions behave differently depending on the direction they are approached. Functions that are differentiable in a neighborhood around a point are called holomorphic and are very smooth.

Differentiable functions on manifolds

See also: Differentiable manifold § Differentiable functions

A differentiable manifold is a special kind of space that looks like regular space when you zoom in close. If we have a function on this space, we say it is differentiable at a point if we can use special maps to check its smoothness around that point. This means the function behaves nicely and can be approximated by straight lines when we look very closely.