Advertisements
Advertisements
Question
Find all points of discontinuity of f, where f is defined by:
f(x) = `{(x/|x|", if" x<0),(-1", if" x >= 0):}`
Advertisements
Solution
f(x) = `{(x/|x|", if" x<0),(-1", if" x >= 0):}`
`lim_(x -> 0^-)` f(x) = `lim_(x -> 0^-) x/abs x`
= `lim_(x -> 0^-) x/(-x)`
= `lim_(x -> 0^-)` (−1)
= −1
`lim_(x -> 0^+)` f(x) = `lim_(x -> 0^+)` (−1) = −1
Also f(0) = −1
Thus, `lim_(x -> 0^-)` f(x) = `lim_(x -> 0^+)` f(x) = f(0)
Hence, f is continuous at x = 0.
APPEARS IN
RELATED QUESTIONS
Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity
`f(x)=(4^x-e^x)/(6^x-1)` for x ≠ 0
`=log(2/3) ` for x=0
Show that the function `f(x)=|x-3|,x in R` is continuous but not differentiable at x = 3.
Find the values of p and q for which
f(x) = `{((1-sin^3x)/(3cos^2x),`
is continuous at x = π/2.
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = –3 and at x = 5.
Find all points 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),(0", if" x = 0):}`
Find all points of discontinuity of f, where f is defined by:
f(x) = `{(x+1", if" x>=1),(x^2+1", if" x < 1):}`
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):}`
Is the function defined by f(x) = `{(x+5", if" x <= 1),(x -5", if" x > 1):}` a continuous function?
Determine if f defined by f(x) = `{(x^2 sin 1/x", if" x != 0),(0", if" x = 0):}` is a continuous function?
Examine the continuity of f, where f is defined by:
f(x) = `{(sin x - cos x", if" x != 0),(-1", if" x = 0):}`
Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.
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: \[f\left( x \right) = \begin{cases}x^{10} - 1, & \text{ if } x \leq 1 \\ x^2 , & \text{ if } x > 1\end{cases}\]
Discuss the Continuity of the F(X) at the Indicated Points : F(X) = | X − 1 | + | X + 1 | at X = −1, 1.
Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.
`lim_("x" -> pi/2)` [sinx] is equal to ____________.
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 function `f(x) = (x^2 - 25)/(x + 5)` is continuous at x =
The point of discountinuity of the function `f(x) = {{:(2x + 3",", x ≤ 2),(2x - 3",", x > 2):}` is are
`f(x) = {{:(x^10 - 1",", if x ≤ 1),(x^2",", if x > 1):}` is discontinuous at
Sin |x| is a continuous function for
Let a, b ∈ R, b ≠ 0. Define a function
F(x) = `{{:(asin π/2(x - 1)",", "for" x ≤ 0),((tan2x - sin2x)/(bx^3)",", "for" x > 0):}`
If f is continuous at x = 0, then 10 – ab is equal to ______.
If functions g and h are defined as
g(x) = `{{:(x^2 + 1, x∈Q),(px^2, x\cancel(∈)Q):}`
and h(x) = `{{:(px, x∈Q),(2x + q, x\cancel(∈)Q):}`
If (g + h)(x) is continuous at x = 1 and x = 3, then 3p + q is ______.
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):}`
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?
