Advertisements
Advertisements
Question
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(3", if" 0 <= x <= 1),(4", if" 1 < x < 3),(5", if" 3 <= x <= 10):}`
Advertisements
Solution
f(x) = `{(3", if" 0 <= x <= 1),(4", if" 1 < x < 3),(5", if" 3 <= x <= 10):}`
For 0 ≤ x ≤ 1, f(x) = 3;
1 < x < 3, f(x) = 4 and
3 ≤ x ≤ 10, f(x) = 5, is a continuous function.
So this is a function.
At x = 1,
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (3) = 3
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (4) = 4
Hence, f is not continuous at x = 1.
At x = 3,
`lim_(x -> 3^-)` f(x) = `lim_(x -> 3^-)` (4) = 4
`lim_(x -> 3^+)` f(x) = `lim_(x -> 3^+)` (5) = 5
Hence, f is not continuous at x = 3.
APPEARS IN
RELATED QUESTIONS
Examine the continuity of the following function :
`{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
Examine the following function for continuity:
f(x) = |x – 5|
Show that
Discuss the continuity of the following functions at the indicated point(s): (iv) \[f\left( x \right) = \left\{ \begin{array}{l}\frac{e^x - 1}{\log(1 + 2x)}, if & x \neq a \\ 7 , if & x = 0\end{array}at x = 0 \right.\]
Discuss the continuity of the following functions at the indicated point(s):
If \[f\left( x \right) = \begin{cases}\frac{x - 4}{\left| x - 4 \right|} + a, \text{ if } & x < 4 \\ a + b , \text{ if } & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, \text{ if } & x > 4\end{cases}\] is continuous at x = 4, find a, b.
If \[f\left( x \right) = \begin{cases}\frac{2^{x + 2} - 16}{4^x - 16}, \text{ if } & x \neq 2 \\ k , \text{ if } & x = 2\end{cases}\] is continuous at x = 2, find k.
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; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
Let\[f\left( x \right) = \left\{ \begin{array}\frac{1 - \sin^3 x}{3 \cos^2 x} , & \text{ if } x < \frac{\pi}{2} \\ a , & \text{ if } x = \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x )^2}, & \text{ if } x > \frac{\pi}{2}\end{array} . \right.\] ]If f(x) is continuous at x = \[\frac{\pi}{2}\] , find a and b.
The function
If \[f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}\]is continuous at x = 0, 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
The value of k which makes \[f\left( x \right) = \begin{cases}\sin\frac{1}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] continuous at x = 0, is
If \[f\left( x \right) = \begin{cases}\frac{\sin \left( \cos x \right) - \cos x}{\left( \pi - 2x \right)^2}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}\]is continuous at x = π/2, then k is equal to
Show that f(x) = x1/3 is not differentiable at x = 0.
Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
The function f (x) = e−|x| is
If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]
If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\] then f (x) is
Find whether the following function is differentiable at x = 1 and x = 2 or not : \[f\left( x \right) = \begin{cases}x, & & x < 1 \\ 2 - x, & & 1 \leq x \leq 2 \\ - 2 + 3x - x^2 , & & x > 2\end{cases}\] .
Examine the continuity of f(x)=`x^2-x+9 "for" x<=3`
=`4x+3 "for" x>3, "at" x=3`
Discuss the continuity of the function f at x = 0
If f(x) = `(2^(3x) - 1)/tanx`, for x ≠ 0
= 1, for x = 0
If f is continuous at x = 0, then find f (0).
Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`
Examine the continuity of the following function :
f(x) = x2 - x + 9, for x ≤ 3
= 4x + 3, for x > 3
at x = 3.
If y = ( sin x )x , Find `dy/dx`
Discuss the continuity of function f at x = 0.
Where f(X) = `[ [sqrt ( 4 + x ) - 2 ]/ ( 3x )]`, For x ≠ 0
= `1/12`, For x = 0
The probability distribution function of continuous random variable X is given by
f( x ) = `x/4`, 0 < x < 2
= 0, Otherwise
Find P( x ≤ 1)
If the function f is continuous at x = I, then find f(1), where f(x) = `(x^2 - 3x + 2)/(x - 1),` for x ≠ 1
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
Let f(x) = `{{:((1 - cos 4x)/x^2",", "if" x < 0),("a"",", "if" x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if" x > 0):}`. For what value of a, f is continuous at x = 0?
Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1
f(x) = `{{:(|x - 4|/(2(x - 4))",", "if" x ≠ 4),(0",", "if" x = 4):}` at x = 4
f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",", "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",", "if" 0 ≤ x ≤ 1):}` at x = 0
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.
Examine the differentiability of f, where f is defined by
f(x) = `{{:(x[x]",", "if" 0 ≤ x < 2),((x - 1)x",", "if" 2 ≤ x < 3):}` at x = 2
`lim_("x" -> "x" //4) ("cos x - sin x")/("x"- "x" /4)` is equal to ____________.
