Advertisements
Advertisements
प्रश्न
Find all the points of discontinuity of f defined by `f(x) = |x| - |x + 1|`.
उत्तर
`f(x) = {(-x - [-(x + 1)], if x<-1),(-(x) - (x+1), if -1 <=x<0),(x - (x+1), if x>=0):}`
`{(1, if x<-1),(-2x-1, if -1 <=x<0),(-1, if x>=0):}`
At = -1
`lim_(x->1^-) f(x) = 1`
`lim_(x->1^+) f(x) = lim_(h->0) (-2(-1+h)) = 1`
f (-1) = -2(-1) -1 = 1
Thus, `lim_(x->1^-) f (x) = lim_(x->1^+) f (x) = f (-1)`
= f is continuous at x = -1
At x= 0
`lim_(x->0^-) f(x) = lim_(x->0^-)(-2x-1) = lim_(h->0)(-2(-h)-1) = -1`
`lim_(x->0^+) f(x) = -1`
Also, f(0) = -1
Thus,`lim_(x->0^-) f(x) = lim_(x->0^+) f(x) = f(0)`
f is continuous at x = 0
Also, f being a constant is continuous when x<-1 or when x>0.
∴ f is continuous for all x ∈ R
Hence, there is no point in discontintinuty.
APPEARS IN
संबंधित प्रश्न
Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Examine the continuity of the function f(x) = 2x2 – 1 at x = 3.
Find all point of discontinuity of f, where f is defined by `f (x) = {(2x + 3, if x<=2),(2x - 3, if x > 2):}`
Find all points of discontinuity of f, where f is defined by `f (x) = {(x/|x|, if x<0),(-1, if x >= 0):}`
Find all points of discontinuity of f, where f is defined by `f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`
Show that the function defined by g(x) = x = [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
Determine if f defined by `f(x) = {(x^2 sin 1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?
Determine the value of the constant 'k' so that function f(x) `{((kx)/|x|, ","if x < 0),(3"," , if x >= 0):}` is continuous at x = 0
Find the value of constant ‘k’ so that the function f (x) defined as
f(x) = `{((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}`
is continous at x = -1
Test the continuity of the function on f(x) at the origin:
\[f\left( x \right) = \begin{cases}\frac{x}{\left| x \right|}, & x \neq 0 \\ 1 , & x = 0\end{cases}\]
Prove that the function
For what value of λ is the function
\[f\left( x \right) = \begin{cases}\lambda( x^2 - 2x), & \text{ if } x \leq 0 \\ 4x + 1 , & \text{ if } x > 0\end{cases}\]continuous at x = 0? What about continuity at x = ± 1?
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}2x , & \text{ if } & x < 0 \\ 0 , & \text{ if } & 0 \leq x \leq 1 \\ 4x , & \text{ if } & x > 1\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}x^{10} - 1, & \text{ if } x \leq 1 \\ x^2 , & \text{ if } x > 1\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou:
Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.
Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`
If f(x) = `{{:("a"x + 1, "if" x ≥ 1),(x + 2, "if" x < 1):}` is continuous, then a should be equal to ______.
`lim_("x" -> pi/2)` [sinx] is equal to ____________.
The number of discontinuous functions y(x) on [-2, 2] satisfying x2 + y2 = 4 is ____________.
Let f (x) `= (1 - "tan x")/(4"x" - pi), "x" ne pi/4, "x" in (0, pi/2).` If f(x) is continuous in `(0, pi/2), "then f"(pi/4) =` ____________.
If f(x) `= sqrt(4 + "x" - 2)/"x", "x" ne 0` be continuous at x = 0, then f(0) = ____________.
`lim_("x"-> 0) sqrt(1/2 (1 - "cos" 2"x"))/"x"` is equal to
The point of discountinuity of the function `f(x) = {{:(2x + 3",", x ≤ 2),(2x - 3",", x > 2):}` is are
How many point of discontinuity for the following function in its. domain.
`f(x) = {{:(x/|x|",", if x < 0),(-1",", if x ≥ 0):}`
How many point of discontinuity for the following function for x ∈ R
`f(x) = {{:(x + 1",", if x ≥ 1),(x^2 + 1",", if x < 1):}`
Sin |x| is a continuous function for
If function f(x) = `{{:((asinx + btanx - 3x)/x^3,",", x ≠ 0),(0,",", x = 0):}` is continuous at x = 0 then (a2 + b2) is equal to ______.
Let α ∈ R be such that the function
f(x) = `{{:((cos^-1(1 - {x}^2)sin^-1(1 - {x}))/({x} - {x}^3)",", x ≠ 0),(α",", x = 0):}`
is continuous at x = 0, where {x} = x – [x], [x] is the greatest integer less than or equal to x.
Find the value(s) of 'λ' if the function
f(x) = `{{:((sin^2 λx)/x^2",", if x ≠ 0 "is continuous at" x = 0.),(1",", if x = 0):}`
Find the value of k for which the function f given as
f(x) =`{{:((1 - cosx)/(2x^2)",", if x ≠ 0),( k",", if x = 0 ):}`
is continuous at x = 0.
If f(x) = `{{:((kx)/|x|"," if x < 0),( 3"," if x ≥ 0):}` is continuous at x = 0, then the value of k is ______.
The graph of the function f is shown below.
Of the following options, at what values of x is the function f NOT differentiable?
