Continuity and Uniform Continuity Real Analysis Lab LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEmcGxvdHNGJ0Y2RjlGQEYrRis=

Continuity: A function f is continuous at c if for each positive number \316\265 there is a positive number \316\264 such that 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 for all x such that | x - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== | < \316\264. Here the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= chosen depends on both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== and the point c. Uniform Continuity: A function is uniformly continuous on a set D if for each positive number \316\265, there is a positive number \316\264 such that |f(u) - f(v)| < e for all x such that | u - v | < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= and u, v contained in D Any uniformly continuous function is also continuous -- but the converse is not true.

Let f(x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw== for 0 \342\211\244 x \342\211\244 20. Define and plot this function. PkkiZkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCokOSQiIiRGJEYkRiQ= QyQ+SSNwMUc2Ii1JJXBsb3RHRiU2JC1JImZHRiU2I0kieEdGJS9GLDsiIiEiIz8iIiI=SSIlRzYi Suppose we are challenged with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== = 0.001 We want to find a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= such that | f(u) - f(v) |< \316\265 =0.001whenever | u - v | < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=. By examing the plot of f(x) it seems that the largest difference | f(u) - f(v) | occurs for u and v near 20. If we can achieve our goal near for u and v near 20, it should hold elsewhere also. Pkkidkc2IiIjPw== LUkiZkc2IjYjSSJ2R0Yk Experiment with different values of \316\264 and u = v - \316\264 = 20 - \316\264 until we get |f(u) - f(v)| < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== = .001. If we try \316\264 = 0.001 to start with it is clearly not small enough: PkkmZGVsdGFHNiIkIiIiISIk PkkidUc2IiwmSSJ2R0YkIiIiSSZkZWx0YUdGJCEiIg== LUkkYWJzRyUqcHJvdGVjdGVkRzYjLCYtSSJmRzYiNiNJInVHRikiIiItRig2I0kidkdGKSEiIg== Once we have found a value for \316\264 that works, verify for a couple of other values of v, say v = 4, and v = 0, that | f(u) - f(v) | < \316\265 = .001 also for |u - v| < \316\264. At v = 4: Pkkidkc2IiIiJQ== PkkidUc2IiwmIiIlIiIiSSZkZWx0YUdGJCEiIg== LCYtSSJmRzYiNiNJInVHRiUiIiItRiQ2I0kidkdGJSEiIg== PkkidUc2IiwmIiIlIiIiSSZkZWx0YUdGJEYn LCYtSSJmRzYiNiNJInVHRiUiIiItRiQ2I0kidkdGJSEiIg== JSFH At v = 0: Pkkidkc2IiIiIQ== PkkidUc2IiwmSSJ2R0YkIiIiSSZkZWx0YUdGJEYn LCYtSSJmRzYiNiNJInVHRiUiIiItRiQ2I0kidkdGJSEiIg==

The function f(x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRy8lKWJldmVsbGVkR1EmZmFsc2VGJw== for x in the open interval ( 0, 1 ) is continuous, but not uniformly continuous. To be continuous at a point x0, it must be true that for any small number \316\265 > 0 with which we are challenged, we can find a \316\264 >0 such that if x is within \316\264 of x0, then f(x) is with \316\265 of f(x0). Pick a point x0. Then pick 3 different small \316\265 (increasingly smaller). Find a corresponding \316\264 for which ( * ) is satisfied. PkkiZ0c2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCokOSQhIiJGJEYkRiQ= LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkiZ0dGJzYjSSJ4R0YnL0YsOyIiISIiIi9JInlHRic7Ri8iIzU= We have continuity at x0 = .02. Pick SShlcHNpbG9uRzYi = .0001. What LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= will work to make |g(x) -g(x0) | < SShlcHNpbG9uRzYi for | x -x0| < \316\264. Try LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUScmIzk0ODtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y3RitGOkY3 = .001. Looking at the graph, we decide that in this interval, |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 | \342\211\244 |g(.019) - g(.02)| LCYtSSJnRzYiNiMkIiM+ISIkIiIiLUYkNiMkIiIjISIjISIi Much too big. Now try \316\264 = .0001; LCYtSSJnRzYiNiMtSSIrRyUqcHJvdGVjdGVkRzYkJCIiIyEiIyQhIiIhIiUiIiItRiQ2I0YrRi8= Still not good enough. Try \316\264 = .00000001. LCYtSSJnRzYiNiMtSSIrRyUqcHJvdGVjdGVkRzYkJCIiIyEiIyQhIiIhIikiIiItRiQ2I0YrRi8= This is plenty small enough for the \316\265 = .0001 challenge. JSFH Why is this function not uniformly continuous? We show this by finding an \316\265 such that no matter how small a \316\264 is chosen, we can always points u and v with | u - v | < \316\264, but g(u) - g(v) > \316\265. From examing the graph, the problem is that the function grows more rapidly as x gets closer to 0. We choose \316\265 = 1. Try any small \316\264 -- .0001 for example. All we have to do is pick the following u and v: PkkidUc2IiwkSSZkZWx0YUdGJCMiIiIiIiM= Pkkidkc2IiwkSSJ1R0YkIyIiIiIiIw== LCYtSSJnRzYiNiNJInVHRiUiIiItRiQ2I0kidkdGJSEiIg== Clearly the absolute value of the above is bigger than \316\265 = 1. What happens if we try again with a smaller \316\264-- say \316\264 = 0.0000001? PkkmZGVsdGFHNiIkIiIiISIo PkkidUc2IiwkSSZkZWx0YUdGJCMiIiIiIiM= Pkkidkc2IiwkSSJ1R0YkIyIiIiIiIw== LCYtSSJnRzYiNiNJInVHRiUiIiItRiQ2I0kidkdGJSEiIg== We definitely have exceeded \316\265 = 1! The problem with the above function is the discontinuity at x = 0. The function is diverging to infinity as x goes to 0. If however we restrict ourselves to functions which are continuous on a closed interval (actually, just need continuous on any compact set) , then we will always have uniform continuity. (See Theorem 23.6, page 218).