Question

Which of the following functions are piecewise continuous on the interval [0, 1] ? 5(x) = x + (0, 1] f(x) = x?, x € [0, 1] x = 0 f(x) = { xe (0, 1] C s(x) = { i x 6 [0, 5) x 6 [.5, 1] 0 x2 f(x) = 1 x 6 [0,.5) x=.5 x 6 (.5, 1] x2

Answer 1

Here we will use the fact that a function is piecewise continious in an interval if it has only finite number of discontinuities in it .

a. f(x) = 1/x , x ∈ (0,1] is piecewise continious in (O,1] as it is continious in whole Interval (0,1] so have no discontinuity in (0,1] .

b. f(x) = x^2 , x∈[0,1] , is also a piecewise continious function as being polynomial it is continious in whole interval [0,1] .

c.f(x) = 0 if x = 0

x^2 if x∈(0,1]

is also picewise continious as only possible point of discontinuity in interval [0,1] is x=0 but here also f(x) is continious and right hand limit have value 0 equals to functional value so it's continious in [0,1] hence piecewise continious .

d. f(x) = 0 , x∈[0,0.5]

= 1 , x∈[0.5,1]

is also piecewise continious as it has only point of discontinuity at x=0.5 in Interval [0,1].

e. f(x) = x^2 , x∈[0,0.5)

= 1 ,x = 0.5

=x^2 ,x∈(0.5,1]

is also piecewise continious as being polynomial it is continious in [0,.5) and (0.5,1] and only point of discontinuity is at x = 0.5 in [0,1] .