0

If a function $f:\mathbb{R}_+ \to \mathbb{R}$ is uniformly continuous and Riemann integrable on $\mathbb{R}_+$ (i.e. $\lim_{x\to\infty} \int_0^x f(s)ds$ exists and is finite), then $f(x)\to 0$ as $x\to\infty$.

Thus, I want to find an example of a continuous function, Riemann integrable on $\mathbb{R}_+$ which does not tend to zero.

Clearly, the limit of $f(x)$ as $x\to\infty$ should not exist.

Martin Sleziak
  • 56,060
Max
  • 31

1 Answers1

0

Let $\psi(x)$ denote the bump function used here, and define $$ I = \int_{-\infty}^{\infty} \psi(x)\, dx.$$ The function $$f(x) = \sum_{n = 1}^\infty \psi(2^n(x-n))$$ is clearly continuous. It is also Riemann integrable, because $$ \lim_{x\to \infty} \int_0^x f(x) \, dx = \sum_{i=1}^\infty 2^{-n} I = I.$$ However $f(x)$ does not converge when $x\to \infty$.

Roberto Rastapopoulos
  • 2,232
  • 13
  • 31