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.
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
so that for each j,
if x and y belong to
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 for all .
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
or suitable constants and b.
By the Lemma, |x| can be approximated by polynomials.
Translating these polynomials by aj shows that we can approximate , and then taking a linear combination shows that we can approximate the piecewise-linear function g,
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