Question

Let f(x)=( sin(1/z), ifrj0 if x = 0. Prove that f(x) has the Intermediate Value Property, although f(r) is not continuous

Answer 1

Solution:

Given that,

Given that, The function f: R rightarrow R defined by f (x) = sin (1/x) if x notequalto 0 and f (0) = 0 Proof for not continuous Consider, lim_x rightarrow 0 f (x) = lim_x rightarrow 0 sin (1/x) (Because x rightarrow 0 Rightarrow x notequalto 0) = sin (1/0) = sin infinity = infinity and f (0) = 0 so, lim_x rightarrow 0 f (x) notequalto f (0) Rightarrow f is not continuous at '0'. The objective is to prove that 'f' has intermediate value property \r\n i.e, for any a < b and f (a) < y f (b) or f (b) < y < f (a) there exists c elementof (a, b) such that f (c) = y This result can be proved as follows Let a < b and lies between f (a), f (b) Case (i): - Let 0 < a < b Suppose g (x) = 1/x, h (x) = sin x Then f (x) = (h o g) (x) = h (g (x)) = h (1/x) sin (1/x) since f is the comportion of continuous functions g and h.