Smooth Function — Definition, Formula & Examples

A smooth function is a function that can be differentiated infinitely many times, with every derivative being continuous. Informally, its graph has no corners, cusps, or breaks of any kind.

A function $f: U \to \mathbb{R}$ , where $U \subseteq \mathbb{R}^n$ is open, is smooth (or $C^\infty$ ) if the partial derivatives $\partial^\alpha f$ exist and are continuous on $U$ for every multi-index $\alpha$ . The class of all such functions on $U$ is denoted $C^\infty(U)$ .

How It Works

To determine whether a function is smooth, you check that its first derivative exists and is continuous, then that its second derivative exists and is continuous, and so on for all orders. A polynomial is always smooth because eventually its higher derivatives become zero. Functions like

e^x

\sin x

, and

\cos x

are smooth on all of

\mathbb{R}

. By contrast,

f(x) = |x|

is not smooth because its first derivative is undefined at

x = 0

Worked Example

Problem: Determine whether

f(x) = x^4 - 3x^2 + 1

is a smooth function.

Step 1: Compute the first derivative.

f'(x) = 4x^3 - 6x

Step 2: Compute higher-order derivatives until a pattern emerges.

f''(x) = 12x^2 - 6, \quad f'''(x) = 24x, \quad f^{(4)}(x) = 24, \quad f^{(n)}(x) = 0 \text{ for } n \geq 5

Step 3: Every derivative exists and is continuous on all of

\mathbb{R}

(they are all polynomials, and eventually become identically zero).

Answer: Yes,

f(x) = x^4 - 3x^2 + 1

is smooth (

C^\infty

) on

\mathbb{R}

. In fact, every polynomial is smooth.

Why It Matters

Smooth functions are the default setting in differential geometry, where curves and surfaces must be differentiated repeatedly. In physics, smooth models of motion and fields ensure that quantities like velocity, acceleration, and force are all well-defined. Taylor series approximations, a core tool in numerical analysis and engineering, require the function to have derivatives of sufficiently high order — smoothness guarantees all of them exist.

Common Mistakes

Mistake: Assuming that any continuous function is smooth.

Correction: Continuity (

C^0

) only means no breaks in the graph. Smoothness (

C^\infty

) additionally requires that every derivative exists and is continuous. For example,

f(x) = |x|

is continuous everywhere but not even once differentiable at

x = 0