HCU MCA 2013 Paper

(c) Given that, f is a continuous real valued function defined on the closed interval [0, 1] and x, y ∈ [0,1]
Now, from Statement III,
|f(x) - f(y)|≤ E|x - y| ,∀ x, y‌‌‌‌‌‌‌‌‌...(i)
where Eq. (i) is true for real values of x + y. Let x is any real number and y is any arbitrary constant. Then,
‌‌‌‌‌‌‌‌‌|

f(x)−f(y)

x−y

|≤E‌and‌y→x
‌‌‌‌‌‌‌

lim

y→x

f(x)−f(y)

x−y

≤

lim

y→x

E‌‌‌‌‌‌‌‌‌‌‌...(ii)
∴‌‌‌‌‌

lim

y→x

f(x)−f(y)

x−y

=f′(x)
From Eq. (ii)
|f'(x)| = E (constant)