Let f x - Bmatrix a- x-clendBmatrix - a - b - c epsilon - R if f x is discontinuous at x - c- and have a jump at x - c then the value of jump is -A- undefined

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Standard XI

Mathematics

Jump of a Discontinuity

Let f x = Bma...

Question

Let f(x) = ${\begin{matrix} a & x > c b & x \geq c \end{matrix}} : a, b, c ϵ - R$
if f(x) is discontinuous at x = c, and have a jump at x = c then the value of jump is -
undefined

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Solution

It is given that the function is discontinuous at x = c, the jump will be nothing but the difference (modulus) of $∣ ∣ ∣ l i m x \to c^{-} f (x) - l i m x \to c^{+} f (x) ∣ ∣ ∣$
= |b - a|

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Q. Let f(x) =

{\begin{matrix} a & x > c b & x \geq c \end{matrix}} : a, b, c ϵ - R

if f(x) is discontinuous at x = c, and have a jump at x = c then the value of jump is -

Let $f (x) = {\begin{matrix} x [\frac{1}{x}] + x [x] & if x \neq 0 0 & if x = 0 \end{matrix}$ where $[\cdot]$ denotes the greatest integer function, then which of the following is true?

The continuity on an interval has a geometric interpretation. namely, a function f defined on an interval I is continuous on I if its graph has no 'holes' or 'jumps' .f is said to have a removable discontinuity at c if f(x) has a limit at c but lim

{lim}_{x \to c} f (x) \neq f (c)

. If

{lim}_{x \to c +} f (x) a n d {lim}_{x \to c -} f (x)

exist but are not equal then c is called jump discontinuity. If

{lim}_{x \to c +} f (x) a n d {lim}_{x \to c -} f (x)

fail to exist then c is called infinite discontinuity.

g (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x + 7 & x < - 3 | x - 2 | & - 3 \leq x < - 1; x^{2} - 2 x & - 1 \leq x < 3 2 x - 3 & x \geq 3 \end{matrix}

then

lim x \to c f (x) \neq f (c)

lim x \to c + f (x) a n d lim x \to c - f (x)

exist but are not equal then c is called jump discontinuity.

lim x \to c + f (x) a n d lim x \to c - f (x)

fail to exist then c is called infinite discontinuity.

For

f (x) = sin x

x = 0

{lim}_{x \to c} f (x) \neq f (c)

{lim}_{x \to c +} f (x) a n d {lim}_{x \to c -} f (x)

exist but are not equal then c is called jump discontinuity.

{lim}_{x \to c +} f (x) a n d {lim}_{x \to c -} f (x)

fail to exist then c is called infinite discontinuity.

Let

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Let f(x) ={ax>cbx≥c} :a,b,cϵ−R if f(x) is discontinuous at x = c, and have a jump at x = c then the value of jump is -undefined

It is given that the function is discontinuous at x = c, the jump will be nothing but the difference (modulus) of ∣∣∣limx→c−f(x)−limx→c+f(x)∣∣∣ = |b - a|

Let f(x) = ${\begin{matrix} a & x > c b & x \geq c \end{matrix}} : a, b, c ϵ - R$
if f(x) is discontinuous at x = c, and have a jump at x = c then the value of jump is -
undefined

It is given that the function is discontinuous at x = c, the jump will be nothing but the difference (modulus) of $∣ ∣ ∣ l i m x \to c^{-} f (x) - l i m x \to c^{+} f (x) ∣ ∣ ∣$
= |b - a|