# How to tell if a function is continuous in an interval

I'm currently creating a program to plot simple graphs in 2D. In order to draw those properly I have to know if a function graph is continuous between 2 points on the graph.

Assuming the graph for f(x)=tan(x) and two points on the graph P1(1,tan(1)) and P2(2,tan(2)). Is there an algorithmic way to tell if the graph is continuous for 1 <= x <= 2?

thanks

• The famous $\epsilon \text{,} \delta$ method is the algorithm you seek. math.stackexchange.com/search?q=epsilon+delta – Arjang Dec 22 '10 at 10:47
• In floating point arithmetic... it's hard to tell. There is the example $f(x)=3x^2+\frac{\ln((\pi-x)^2)}{\pi^4}+1$. The best you can do is that during the plotting process, ensure that either the slope or distance between two consecutive points on the graph is not "too big". – J. M. is a poor mathematician Dec 22 '10 at 10:48
• You might also want to look into implementing adaptive subdivision to ensure that glaring discontinuities are easily caught. – J. M. is a poor mathematician Dec 22 '10 at 10:59

## 1 Answer

You can use interval arithmetic to compute reliable plots. See for instance this paper: Jeff Tupper, Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables, SIGGRAPH 2001. The excellent GrafEq software uses this technique.

• Interval arithmetic is the "guaranteed correct" way of catching discontinuities in a plot; it's not (very) fast, but if the functions to be plotted are simple enough, it's very reliable. – J. M. is a poor mathematician Dec 22 '10 at 11:09