My English lesson The continuity of a function is thus a necessary, but not sufficient, condition for its derivability. The set of derivable functions is a proper subset of the set of continuous functions. BE CAREFUL! B D Derivable continuous hence non continuous non differentiable derivable functions continuous functions real functions From the graph of the function y = |x|, we can construct graphs of functions with infinite points of non-derivability. Let us consider the function y = 2 |x| in the interval [ 1 ; +1] and break its graph so that it has three angular points and we repeat this procedure of breaking up, ad infinitum. y y 1 2 O 1 1 y O 1 O 1 x y O 1x 1 x y x O NOW IT S YOUR TURN N x B calculating the derivative at a By generic point x = x0 as the limit of the incremental ratio of the functions in the example considered: 2x 5 a. y = _ 3 x b. y = | x2 3x 4| We thus obtain an example of a function that is always continuous in an interval but non-derivable at infinite points. y example 1 O Roughly graph the following functions and determine the points at which they are non-derivable: 2x 5 3 x a. y = _ O 1 2 3 x 2 b. y = | x2 3x 4| a. The function is not defined for x = 3; its graph is a hyperbola that has as its vertical asymptote the line x = 3 and as its horizontal asymptote the line y = 2. Only at x = 3, where it is not defined, is the function not continuous and therefore non-derivable. b. The function y = |x2 3x 4| is defined and continuous at every x R. The function is non-derivable at x = 1 and at x = 4 where it has two angular points. By calculating the derivative at a generic point x = x0 as a limit of the incremental ratio we get: 2x 3 if x 1 o x 4 f (x) = therefore { 2x + 3 if 1 < x < 4 lim f (x) = 5 and x 1 lim f (x) = 5 and x 4 a. y lim f (x) = 5 x 1+ 1 lim f (x) = 5 x 4+ b. O 1 x 277

My English lesson