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 $\mathrm{III}$,
|f(x) - f(y)|≤ E|x - y| ,∀ x, y$...\left(\mathrm{i}\right)$
where Eq. $\left(\mathrm{i}\right)$ is true for real values of x + y. Let x is any real number and y is any arbitrary constant. Then,
$|\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(\mathrm{y}\right)}{\mathrm{x}-\mathrm{y}}|\le \mathrm{E}\mathrm{and}\mathrm{y}\to \mathrm{x}$
$\underset{\mathrm{y}\to \mathrm{x}}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(\mathrm{y}\right)}{\mathrm{x}-\mathrm{y}}\le \underset{\mathrm{y}\to \mathrm{x}}{\lim }\mathrm{E}...\left(\mathrm{ii}\right)$
∴$\underset{\mathrm{y}\to \mathrm{x}}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(\mathrm{y}\right)}{\mathrm{x}-\mathrm{y}}=\mathrm{f}\prime \left(\mathrm{x}\right)$
From Eq. $\left(\mathrm{ii}\right)$
|f'(x)| = E (constant)