Advertisements
Advertisements
Question
Examine the continuity of f, where f is defined by:
f(x) = `{(sin x - cos x", if" x != 0),(-1", if" x = 0):}`
Advertisements
Solution
f(x) = `{(sin x - cos x", if" x != 0),(-1", if" x = 0):}`
Approach 1:
If f(x) is continuous at x = c, it implies:
f(c) = `lim_(x -> c^+)` f(x) = `lim_(x -> c^-)` f(x)
⇒ −1 = sin 0 − cos 0 = −sin 0 − cos 0
⇒ −1 = −1 = −1
This shows that f(x) is continuous at x = 0.
Approach 2:
c ≠ 0 and c ⊂ R
If f(x) is continuous at x = c, it implies:
sin c − cos c is continuous, i.e. sin c and cos c are continuous functions, which is true.
That is, f(x) is also 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
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.
Examine the continuity of the function f(x) = 2x2 – 1 at x = 3.
Prove that the function f(x) = xn is continuous at x = n, where n is a positive integer.
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/|x|", if" x<0),(-1", 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):}`
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?
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
Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.
Test the continuity of the function on f(x) at the origin:
\[f\left( x \right) = \begin{cases}\frac{x}{\left| x \right|}, & x \neq 0 \\ 1 , & x = 0\end{cases}\]
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 relationship between 'a' and 'b' so that the function 'f' defined by
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: \[f\left( x \right) = \begin{cases}x^{10} - 1, & \text{ if } x \leq 1 \\ x^2 , & \text{ if } x > 1\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}- 2 , & \text{ if }& x \leq - 1 \\ 2x , & \text{ if } & - 1 < x < 1 \\ 2 , & \text{ if } & x \geq 1\end{cases}\]
Find the point of discontinuity, if any, of the following function: \[f\left( x \right) = \begin{cases}\sin x - \cos x , & \text{ if } x \neq 0 \\ - 1 , & \text{ if } x = 0\end{cases}\]
Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.
Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`
`lim_("x" -> pi/2)` [sinx] is equal to ____________.
The number of discontinuous functions y(x) on [-2, 2] satisfying x2 + y2 = 4 is ____________.
If f(x) `= sqrt(4 + "x" - 2)/"x", "x" ne 0` be continuous at x = 0, then f(0) = ____________.
The function `f(x) = (x^2 - 25)/(x + 5)` is continuous at x =
The function f defined by `f(x) = {{:(x, "if" x ≤ 1),(5, "if" x > 1):}` discontinuous at x equal to
The point of discountinuity of the function `f(x) = {{:(2x + 3",", x ≤ 2),(2x - 3",", x > 2):}` is are
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 ______.
If the function f defined as f(x) = `1/x - (k - 1)/(e^(2x) - 1)` x ≠ 0, is continuous at x = 0, then the ordered pair (k, f(0)) is equal to ______.
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?
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?
