Question:

The function $f(x) = [x]$, where $[x]$ denotes greatest integer function is continuous at

Updated On: Apr 2, 2024

4
-2
1
1.5

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The Correct Option is D

Solution and Explanation

$[x]$ is always continuous at non-integer value of $x$. Hence, $f(x)=[x]$ will be continuous at $x=1.5$.

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Concepts Used:

Differentiability

Differentiability of a function A function f(x) is said to be differentiable at a point of its domain if it has a finite derivative at that point. Thus f(x) is differentiable at x = a
$\frac{d y}{d x}=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h}=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$
⇒ f'(a – 0) = f'(a + 0)
⇒ left-hand derivative = right-hand derivative.
Thus function f is said to be differentiable if left hand derivative & right hand derivative both exist finitely and are equal.
If f(x) is differentiable then its graph must be smooth i.e. there should be no break or corner.
Note:
(i) Every differentiable function is necessarily continuous but every continuous function is not necessarily differentiable i.e. Differentiability ⇒ continuity but continuity ⇏ differentiability

(ii) For any curve y = f(x), if at any point $\frac{d y}{d x}$ = 0 or does not exist then, the point is called “critical point”.

3. Differentiability in an interval
(a) A function fx) is said to be differentiable in an open interval (a, b), if it is differentiable at every point of the interval.

(b) A function f(x) is differentiable in a closed interval [a, b] if it is

Differentiable at every point of interval (a, b)
Right derivative exists at x = a
Left derivative exists at x = b.