My English lesson The definition of continuity We can intuitively identify the continuity of the graph of a function and, therefore, of the function itself with the possibility of drawing it without lifting the pen tip from the paper. To make this concept rigorous we need a formal definition. DEFINITION A function y = f(x) is continuous at a point a R if it is defined at the point a and its limit, as x tends to a, coincides with the value of the function at a. In symbols: lim f(x) = f(a) x a From this, we can deduce that in cases where these conditions are not fulfilled we will speak of discontinuity of a function at a point. In particular (see graphs below): I. if it is not defined at x = a, we can have a vertical asymptote or a hole; II. if it is defined at a but lim f(x) lim f(x) the graph of the function has a jump; x a x a+ III. if it is defined at a and lim f(x) = lim+ f(x) but lim f(x) f(a), the value of the x a x a x a function indicated by the point (a ; f(a)), is different from the limit to which the function tends from both left and right. y O y a x O y a O x I. y a O x II. a x III. The third is a special case: in fact, it is possible to eliminate the point of discontinuity by redefining the function by cases to make it continuous over the whole set R: y= f(x) if x a f(x) {lim x a if x = a If, therefore, a function is not defined at a single point, it may be interesting to establish whether it exists and what its limit is at that the point; its discontinuity might in fact be eliminated. example O Calculate for what value of m the following function defined by cases is continuous in R: x2 x LI x 1, the function y = mx represents, as m varies, a pencil of lines passing through the origin. The only point of discontinuity could be that of abscissa x = 1 (fig. a. next page). y= For it to be continuous even at that the point, it must be: lim y = lim y y lim x2 x = 4 x 1 x 1+ x 1 and lim mx + 3 = m + 3 4 = m + 3 m = 7 x 3+ 170

My English lesson