Advertisements
Advertisements
Question
Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.
Solution
Continuity at x = 1
`(x = 1)= x^2 = (1)^2 = 1`
`lim_(x->1^+) f(x) = lim_(x->1^+) 1/x = 1`
`lim_(x->1^-) f(x) = lim_(x->1^-) x^2 = 1`
`:. f(x = 1) = lim_(x->1^-) f(x) = lim_(x->1^+) f(x) = 1`
:. f(x) is continuous at x = 1
Now differentiable at x = 1
(R.H.D at x = 1) = `lim_(x->1^+) (f(x) - f(1))/(x -1)`
`=lim_(x->1) (1/x- 1)/(x - 1)`
`= lim_(x-> 1) (-(x-1) 1/x)/(x-1)`
`= = -1/1 = -1`
(L.H.D at x = 1) = `lim_(x -> 1^(-)) (f(x)-f(1))/(x-1)`
`= lim_(x->1^1) (x^2 -1)/(x - 1) = 2`
`:. l.H.D != R.H.D`
:. f(x) is not differentiable at x = 1
APPEARS IN
RELATED QUESTIONS
Find the values of p and q for which
f(x) = `{((1-sin^3x)/(3cos^2x),`
is continuous at x = π/2.
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|+3, if x<= -3),(-2x, if -3 < x < 3),(6x + 2, if x >= 3):}`
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):}`
Find the points of discontinuity of f, where `f (x) = {(sinx/x, if x<0),(x + 1, if x >= 0):}`
Determine if f defined by `f(x) = {(x^2 sin 1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?
Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.
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
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: \[f\left( x \right) = \begin{cases}2x , & \text{ if } & x < 0 \\ 0 , & \text{ if } & 0 \leq x \leq 1 \\ 4x , & \text{ if } & x > 1\end{cases}\]
If f(x) = `{{:("a"x + 1, "if" x ≥ 1),(x + 2, "if" x < 1):}` is continuous, then a should be equal to ______.
Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`
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) =` ____________.
`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):}`
`f(x) = {{:(x^10 - 1",", if x ≤ 1),(x^2",", if x > 1):}` is discontinuous at
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 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 ______.
Consider the graph `y = x^(1/3)`
Statement 1: The above graph is continuous at x = 0
Statement 2: The above graph is differentiable at x = 0
Which of the following is correct?
