Advertisements
Advertisements
Question
Show that the function `f(x)=|x-3|,x in R` is continuous but not differentiable at x = 3.
Advertisements
Solution
`f(x)=|x-3|={(3-x, x<3),(x-3,x>=3) :}`
Let c be a real number.
Case I: c < 3. Then f (c) = 3 − c.
`lim_(x->c)f(x)=lim_(x->c)(3-x)=3-c`
Since `lim_(x->c)f(x)=f(c)` f is continuous at all negative real numbers.
Case II: c = 3. Then f (c) = 3 − 3 = 0
`lim_(x->c)f(x)=lim_(x->c)(x-3)=3-3=0`
Since , `lim_(x->3)f(x)=f(3)` ,f is continuous at x = 3.
Case III: c > 3. Then f (c) = c − 3.
`lim_(x->c)f(x)=lim_(x->c)(x-3)=c-3`
Since `lim_(x->c)f(x)=f(c)` f is continuous at all positive real numbers.
Therefore, f is continuous function.
We will now show that f(x)=|x-3|,x in R is not differentiable at x = 3.
Consider the left hand limit of f at x = 3
`lim_(h->0^-)(f(3+h)-f(3))/h=lim_(h->0^-)(|3+h-3|-|3-3|)/h=lim_(h->0^-)(|h|-0)/h=lim_(h->0^-)-h/h=-1`
consider the right hand limit of f at x=3
`lim_(h->0^-)(f(3+h)-f(3))/h=lim_(h->0^-)(|3+h-3|-|3-3|)/h=lim_(h->0^-)(|h|-0)/h=lim_(h->0^-)h/h=-1`
Since the left and right hand limits are not equal, f is not differentiable at x = 3.
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.
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) = `{(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|/x", if" x != 0),(0", if" x = 0):}`
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):}`
Is the function defined by f(x) = `{(x+5", if" x <= 1),(x -5", if" x > 1):}` a continuous function?
Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.
Determine the value of the constant 'k' so that function f(x) `{((kx)/|x|, ","if x < 0),(3"," , if x >= 0):}` is continuous at x = 0
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}2x , & \text{ if } & x < 0 \\ 0 , & \text{ if } & 0 \leq x \leq 1 \\ 4x , & \text{ if } & x > 1\end{cases}\]
The function f (x) = tan x is discontinuous on the set
Discuss the Continuity of the F(X) at the Indicated Points : F(X) = | X − 1 | + | X + 1 | at X = −1, 1.
Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.
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)`
`lim_("x" -> pi/2)` [sinx] is equal to ____________.
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) =` ____________.
The function f defined by `f(x) = {{:(x, "if" x ≤ 1),(5, "if" x > 1):}` discontinuous at x equal to
`f(x) = {{:(x^3 - 3",", if x < 2),(x^2 + 1",", if x > 2):}` has how many point of discontinuity
`f(x) = {{:(x^10 - 1",", if x ≤ 1),(x^2",", if x > 1):}` is discontinuous at
Sin |x| is a continuous function for
If f(x) = `{{:((log_(sin|x|) cos^2x)/(log_(sin|3x|) cos x/2), |x| < π/3; x ≠ 0),(k, x = 0):}`, then value of k for which f(x) is continuous at x = 0 is ______.
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 ______.
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?
