Advertisements
Advertisements
Question
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(2x", if" x < 0),(0", if" 0 <= x <= 1),(4x", if" x > 1):}`
Advertisements
Solution
f(x) = `{(2x", if" x < 0),(0", if" 0 <= x <= 1),(4x", if" x > 1):}`
For x < 0, f(x) = 2x;
0 < x < 1, f(x) = 0 and
for x > 1, f(x) = 4x is a polynomial and continuous function.
So this is a function.
At x = 0,
`lim_(x -> 0^-)` f(x) = `lim_(x -> 0^-)` (2x)
= `lim_(h -> 0)` [2(0 − h)]
= `lim_(h -> 0)` (−2h)
= −2 × 0
= 0
`lim_(x -> 0^+)` f(x) = `lim_(x -> 0^+)` (0) = 0
Hence, f is continuous at x = 0.
At x = 1,
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (0) = 0
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (4x)
= `lim_(h -> 0)` [4(1 + h)]
= `lim_(h -> 0)` (4 + 4h)
= 4 + 4 × 0
= 4
Hence, it is not continuous at x = 1.
APPEARS IN
RELATED QUESTIONS
Find the value of 'k' if the function
`f(X)=(tan7x)/(2x) , "for " x != 0 `
`=k`, for x=0
is continuos at x=0
Let \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x^2}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\] Show that f(x) is discontinuous at x = 0.
Discuss the continuity of the following function at the indicated point:
`f(x) = {{:(|x| cos (1/x)",", x ≠ 0),(0",", x = 0):} at x = 0`
Discuss the continuity of the following functions at the indicated point(s):
Find the value of k for which \[f\left( x \right) = \begin{cases}\frac{1 - \cos 4x}{8 x^2}, \text{ when} & x \neq 0 \\ k ,\text{ when } & x = 0\end{cases}\] is continuous at x = 0;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}\]at x = 1
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 + x^3 + 2 x^2}{\tan^{- 1} x}, & \text{ if } x \neq 0 \\ 10 , & \text{ if } x = 0\end{cases}\]
The function
If the function \[f\left( x \right) = \begin{cases}\left( \cos x \right)^{1/x} , & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then the value of k is
Let f (x) = | x | + | x − 1|, then
If \[f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.\] then the value of (a, b) for which f (x) cannot be continuous at x = 1, is
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
If \[f\left( x \right) = \begin{cases}a x^2 - b, & \text { if }\left| x \right| < 1 \\ \frac{1}{\left| x \right|} , & \text { if }\left| x \right| \geq 1\end{cases}\] is differentiable at x = 1, find a, b.
Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.
Is every continuous function differentiable?
Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
If \[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . ,\]
then at x = 0, f (x)
The function f (x) = |cos x| is
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
Find the value of k for which the function f (x ) = \[\binom{\frac{x^2 + 3x - 10}{x - 2}, x \neq 2}{ k , x^2 }\] is continuous at x = 2 .
Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`
The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.
If f(x) = `(e^(2x) - 1)/(ax)` . for x < 0 , a ≠ 0
= 1. for x = 0
= `(log(1 + 7x))/(bx)`. for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b
Find the value of 'k' if the function
f(x) = `(tan 7x)/(2x)`, for x ≠ 0.
= k for x = 0.
is continuous at x = 0.
If the function f is continuous at x = 2, then find 'k' where
f(x) = `(x^2 + 5)/(x - 1),` for 1< x ≤ 2
= kx + 1 , for x > 2
If the function
f(x) = x2 + ax + b, x < 2
= 3x + 2, 2≤ x ≤ 4
= 2ax + 5b, 4 < x
is continuous at x = 2 and x = 4, then find the values of a and b
The function f(x) = |x| + |x – 1| is ______.
The set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.
For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).
f(x) = `{{:((1 - cos 2x)/x^2",", "if" x ≠ 0),(5",", "if" x = 0):}` at x = 0
f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1
Prove that the function f defined by
f(x) = `{{:(x/(|x| + 2x^2)",", x ≠ 0),("k", x = 0):}`
remains discontinuous at x = 0, regardless the choice of k.
Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",", "if" x ≤ 1),("q"x + 2",", "if" x > 1):}` is differentiable at x = 1
If f is continuous on its domain D, then |f| is also continuous on D.
Write the number of points where f(x) = |x + 2| + |x - 3| is not differentiable.
If the following function is continuous at x = 2 then the value of k will be ______.
f(x) = `{{:(2x + 1",", if x < 2),( k",", if x = 2),(3x - 1",", if x > 2):}`
