Advertisements
Advertisements
Question
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
Options
continuous everywhere
continuous at integer points only
continuous at non-integer points only
differentiable everywhere
Advertisements
Solution
(c) continuous at non-integer points only
\[\text{ We have }, \]
\[f\left( x \right) = x - \left[ x \right]\]
\[\text{Consider n be an integer} . \]
`f(x) = x - [x] = {(x-(n-1),n-1le x <n),(0, x = n),(x-n , n le x < n +1):}`
Now,
\[\left( \text { LHL at x = n } \right) = \lim_{x \to n^-} f\left( x \right) = x - \left( n - 1 \right) = x - n + 1\]
\[\left( \text { RHL at x } = n \right) = \lim_{x \to n^+} f\left( x \right) = x - \left( n \right) = x - n\]
\[\text { As, LHL } \neq\text { RHL at x } = n\]
\[i . e . , \text{given function is not continuous at n} . \]
\[\text{Now, n is any integer} . \]
\[\text{Therefore, given function is not continuous at integers} . \]
Therefore, given points are continuous at non-integer points only.
APPEARS IN
RELATED QUESTIONS
A function f (x) is defined as
f (x) = x + a, x < 0
= x, 0 ≤x ≤ 1
= b- x, x ≥1
is continuous in its domain.
Find a + b.
For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if" x <= 0),(4x+ 1", if" x > 0):}` continuous at x = 0? What about continuity at x = 1?
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{((kcosx)/(pi-2x)", if" x != pi/2),(3", if" x = pi/2):}` at x = `"pi/2`
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx +1", if" x<= pi),(cos x", if" x > pi):}` at x = π
Find the values of a and b such that the function defined by f(x) = `{(5", if" x <= 2),(ax +b", if" 2 < x < 10),(21", if" x >= 10):}` is a continuous function.
Let \[f\left( x \right) = \frac{\log\left( 1 + \frac{x}{a} \right) - \log\left( 1 - \frac{x}{b} \right)}{x}\] x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
If \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin } x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (0).
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2kx}{x^2}, \text{ if } & x \neq 0 \\ 8 , \text{ if } & x = 0\end{cases}\] at x = 0
Find the values of a and b so that the function f given by \[f\left( x \right) = \begin{cases}1 , & \text{ if } x \leq 3 \\ ax + b , & \text{ if } 3 < x < 5 \\ 7 , & \text{ if } x \geq 5\end{cases}\] is continuous at x = 3 and x = 5.
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x
Show that f (x) = cos x2 is a continuous function.
Determine the value of the constant 'k' so that function f
\[f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}\] then f (x) is continuous for all
If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to
If \[f\left( x \right) = \begin{cases}\frac{\log\left( 1 + ax \right) - \log\left( 1 - bx \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] and f (x) is continuous at x = 0, then the value of k is
The function
The function
If the function f (x) defined by \[f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then k =
Find the values of a and b so that the function
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
Let f (x) = |cos x|. Then,
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
If f.g is continuous at x = a, then f and g are separately continuous at x = a.
`lim_("x" -> 0) ("x cos x" - "log" (1 + "x"))/"x"^2` is equal to ____________.
The point(s), at which the function f given by f(x) = `{("x"/|"x"|"," "x" < 0),(-1"," "x" ≥ 0):}` is continuous, is/are:
The function f(x) = x2 – sin x + 5 is continuous at x =
Which concept is fundamental in determining continuity of a function at a point?
If \[f\] and \[g\] are continuous at \[x=c\], which collection is continuous at \[x=c\]?
For \[\frac{f}{g}\] to be continuous at \[x=c\], which condition is required?
If \[g\] is continuous and \[g(x)\neq 0\], which function is continuous?
For \[f(x)=\frac{p(x)}{q(x)}\], where \[p(x)\] and \[q(x)\] are polynomial functions, where is \[f\] continuous?
At which values is \[\tan x=\dfrac{\sin x}{\cos x}\] not continuous?
Why is \[f(x)=\sin(x^2)\] continuous for all real \[x\]?
Why is \[g(x)=1-x+|x|\] continuous?
