## You are here

Homeone-sided continuity by series

## Primary tabs

# one-sided continuity by series

Theorem. If the function series

$\displaystyle\sum_{{n=1}}^{\infty}f_{n}(x)$ | (1) |

is uniformly convergent on the interval $[a,\,b]$, on which the terms $f_{n}(x)$ are continuous from the right or from the left, then the sum function $S(x)$ of the series has the same property.

Proof. Suppose that the terms $f_{n}(x)$ are continuous from the right. Let $\varepsilon$ be any positive number and

$S(x)\;:=\;S_{n}(x)+R_{{n+1}}(x),$ |

where $S_{n}(x)$ is the $n^{\mathrm{th}}$ partial sum of (1) ($n\,=\,1,\,2,\,\ldots$). The uniform convergence implies the existence of a number $n_{\varepsilon}$ such that on the whole interval we have

$|R_{{n+1}}(x)|<\frac{\varepsilon}{3}\quad\mathrm{when}\;\;n>n_{\varepsilon}.$ |

Let now $n>n_{\varepsilon}$ and $x_{0},\,x_{0}\!+\!h\in[a,\,b]$ with $h>0$. Since every $f_{n}(x)$ is continuous from the right in $x_{0}$, the same is true for the finite sum $S_{n}(x)$, and therefore there exists a number $\delta_{\varepsilon}$ such that

$|S_{n}(x_{0}\!+\!h)-S_{n}(x_{0})|<\frac{\varepsilon}{3}\quad\mathrm{when}\;\;0% <h<\delta_{\varepsilon}.$ |

Thus we obtain that

$\displaystyle|S(x_{0}\!+\!h)-S(x_{0})|$ | $\displaystyle\;=\;|[S_{n}(x_{0}\!+\!h)-S_{n}(x_{0})]+R_{{n+1}}(x_{0}\!+\!h)-R_% {{n+1}}(x_{0}|$ | ||

$\displaystyle\;\leqq\;|S_{n}(x_{0}\!+\!h)-S_{n}(x_{0})|+|R_{{n+1}}(x_{0}\!+\!h% )|+|R_{{n+1}}(x_{0})|$ | |||

$\displaystyle\;<\;\frac{\varepsilon}{3}\!+\!\frac{\varepsilon}{3}\!+\!\frac{% \varepsilon}{3}\;=\;\varepsilon$ |

as soon as

$0<h<\delta_{\varepsilon}.$ |

This means that $S$ is continuous from the right in an arbitrary point $x_{0}$ of $[a,\,b]$.

Analogously, one can prove the assertion concerning the continuity from the left.

## Mathematics Subject Classification

40A30*no label found*26A03

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections