Question #216577

A function is called “piecewise linear” if it is (i) continuous and (ii) its graph consists of finitely many linear segments. Prove that a continuous function on an interval [a, b] is the uniform limit of a sequence of piecewise linear functions. 


1
Expert's answer
2021-07-13T14:46:57-0400

Answer:

as given,

f is piecewise linear so,

f is continuous on [a, b], it is uniformly continuous.

Thus, given e > 0, we may partition [a, b] by points x0=a,x1,...,xn=b,x_0 = a, x_1, . . . , x_n = b,

so that for each j,

f(x)f(y)<e|f(x) − f(y)| < e if x and y belong to [xj1,xj][x_{j-1}, x_j]

Let g be the function which is linear on each interval [xj-1, xj] and is such that g(xj) = f(xj ) for each j.

Then it is easy to see that f(x)g(x)<2e|f(x) − g(x)| < 2e for all x[a,b]x ∈ [a, b] .

We conclude that f can be uniformly approximated by piecewise-linear functions, so it suffices to show that any piecewise-linear function can be approximated by polynomials.

Now, any piecewise-linear function g can be written as

g(x)=b+j=1ncjxajg(x)=b+\sum_{j=1}^{n}c_j|x-a_j|

or suitable constants n,aj,cjn, a_j , c_j and b.

By the Lemma, |x| can be approximated by polynomials.

Translating these polynomials by aj shows that we can approximate xaj|x−a_j | , and then taking a linear combination shows that we can approximate the piecewise-linear function g,


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