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For a function $f:[0,1]\to \mathbb{R}$ and an integer $n\ge 1$, its Bernstein polynomial isThe Bernstein basis supplies nonnegative weights whose sum is one. Equivalently, the weights are the probabilities of a binomial distribution, and the result is an expected value. At a given $x$, this averages the sampled values $f(k/n)$ with weights concentrated near $k/n=x$.^[ref]

The endpoints are exact: $B_{n}f(0)=f(0)$ and $B_{n}f(1)=f(1)$. Interior sampled points need not be interpolated. For example, binomial distribution moments give $B_n(x)=x$ and $B_n(x^2)=x^2+x(1-x)/n$. Thus every affine function is reproduced exactly, whereas $x^2$ is generally changed. Bernstein polynomial degree preservation is a statement about degree of a polynomial, not equality of the entire polynomial.

For a continuous $f$, the uniform continuity of $f$ and the variance $\mathrm{Var}(K/n)=x(1-x)/n\le 1/(4n)$ explain uniform convergence. Split the expected value of the error into $|K/n-x|<\delta$, where uniform continuity makes the change small, and its complement, whose probability is at most $1/(4n{\delta }^2)$ by Chebyshev's inequality. The bounded function $f$ bounds the remaining error. This proves convergence without assuming differentiability.

For $f\in C([0,1{]}^d)$, the tensor-product Bernstein polynomial is the expected value of $f(X_1/n,\dots ,X_d/n)$ for independent random variables $X_j\sim \mathrm{Bin}(n,x_j)$. The same uniform continuity and coordinatewise variance estimates give uniform convergence for each fixed finite $d$.

Every continuous real-valued function on a compact interval can be approximated by polynomials with arbitrarily small error everywhere on the interval. More precisely, if $a<b$ and $f:[a,b]\to \mathbb{R}$ is continuous, then for every $\epsilon >0$ there is a polynomial $p$ withThe error is measured in the supremum norm, so one approximation works at every point. The degenerate interval $a=b$ needs only a constant function. A change of variable $t=(x-a)/(b-a)$ reduces the nondegenerate case to $[0,1]$.^[ref]

This is useful because a polynomial is described by finitely many coefficients and can be evaluated with addition and multiplication. Its surprise is that the target need not be differentiable: the sharp corner of an absolute value function can be approximated by smooth polynomials. The Bernstein approximation of a corner below makes this visible. Approximating values, however, does not by itself approximate derivatives; Weierstrass approximation does not preserve derivatives gives a counterexample.