Question

Let f : R → R is a real valued function ∀ x, y ∈ R such that | f(x) – f(y)| ≤ |x – y|² . The function h(x) = $\log 2$dx is –

• A. Every where continuous
• B. Discontinuous at x = 0 only
• C. Discontinuous at all integral points
• D. h(0) = 0

Answer 1

Answer: (A)

Every where continuous

Answer 2

Since |f(x) – f(y)| ≤ | x – y |² x ≠ y

∴ $\left| \frac { f ( x ) - f ( y ) } { x - y } \right|$ ≤ | x – y |

Taking lim as y → x, we get

$\lim _ { y \rightarrow x }$ $\left| \frac { f ( x ) - f ( y ) } { x - y } \right|$ ≤ $\lim _ { y \rightarrow x }$ |x – y|

⇒ $\left| \lim _ { y \rightarrow x } \frac { f ( x ) - f ( y ) } { x - y } \right|$ ≤ $\left| \lim _ { y \rightarrow x } ( x - y ) \right|$

⇒ |f ′ (x) ≤ 0| ⇒ |f ′(x)| = 0 (Q |f′(x) ≥ 0|)

∴ f ′(x) = 0 ⇒ f(x) = C (constant)

∴ h(x) = dx =$\int \mathrm { Cdx }$= cx + d  where d is constant of integration.

h(x) of a linear function of x which is continuous for all x ∈ R.