LimitsHard
Question
Let [x] denote the integral part of x ∈ R and g(x) = x - [x]. Let f(x) be any continuous function with f(0) = f(1), then the function h(x) = f(g(x)) :
Options
A.has finitely many discontinuities
B.is continuous on R
C.is discontinuous at some x = c
D.is a constant function.
Solution
g(x) = x - [x] = {x}∈ [0, 1)
g(x) is discontinuous only at x ∈ I
Now h(x) = fog(x)
h(x) is continuous ∀ x ∈ R - l
Let x ∈ I, consider x = n
h(n) = f[g(n)] = f(0)
= f(1) = f(0)
f({x}) = f(0)
⇒ h(x) is continuous ∀ x ∈ l
⇒ h(x) is continuous ∀ x ∈ R
g(x) is discontinuous only at x ∈ I
Now h(x) = fog(x)
h(x) is continuous ∀ x ∈ R - l
Let x ∈ I, consider x = n
h(n) = f[g(n)] = f(0)
= f(1) = f(0)f({x}) = f(0)⇒ h(x) is continuous ∀ x ∈ l
⇒ h(x) is continuous ∀ x ∈ R
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