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Is Every Continuous Function Differentiable? - Mathematics

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प्रश्न

Is every continuous function differentiable?

उत्तर

No, function may be continuous at a point but may not be differentiable at that point .
For example: function  

\[f(x) = |x|\]  is continuous at 
 
\[x = 0\]  but it is not differentiable at 
\[x = 0\]
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पाठ 10: Differentiability - Exercise 10.3 [पृष्ठ १७]

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आरडी शर्मा Mathematics [English] Class 12
पाठ 10 Differentiability
Exercise 10.3 | Q 3 | पृष्ठ १७

व्हिडिओ ट्यूटोरियलVIEW ALL [4]

संबंधित प्रश्‍न

 If 'f' is continuous at x = 0, then find f(0).

`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`


If \[f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}\] find whether f is continuous at x = 0.


Show that

\[f\left( x \right)\] = \begin{cases}\frac{x - \left| x \right|}{2}, when & x \neq 0 \\ 2 , when & x = 0\end{cases}

is discontinuous at x = 0.

 

Discuss the continuity of the following functions at the indicated point(s): 

\[f\left( x \right) = \left\{ \begin{array}{l}(x - a)\sin\left( \frac{1}{x - a} \right), & x \neq a \\ 0 , & x = a\end{array}at x = a \right.\]

 


Discuss the continuity of the following functions at the indicated point(s): 

\[f\left( x \right) = \left\{ \begin{array}{l}\frac{2\left| x \right| + x^2}{x}, & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]

Discuss the continuity of the following functions at the indicated point(s): 

\[f\left( x \right) = \binom{\left| x - a \right|\sin\left( \frac{1}{x - a} \right), for x \neq a}{0, for x = a}at x = a\] 

Discuss the continuity of the function f(x) at the point x = 1/2, where \[f\left( x \right) = \begin{cases}x, 0 \leq x < \frac{1}{2} \\ \frac{1}{2}, x = \frac{1}{2} \\ 1 - x, \frac{1}{2} < x \leq 1\end{cases}\] 


Discuss the continuity of \[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0\]


For what value of k is the function

\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]  continuous at x = 0?

 


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; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\] 

 


Discuss the continuity of the f(x) at the indicated points: 

(i) f(x) = | x | + | x − 1 | at x = 0, 1.


Prove that  \[f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{x}, & x \neq 0 \\ 2 , & x = 0\end{cases}\] is discontinuous at x = 0

 


For what value of k is the following function continuous at x = 2? 

\[f\left( x \right) = \begin{cases}2x + 1 ; & \text{ if } x < 2 \\ k ; & x = 2 \\ 3x - 1 ; & x > 2\end{cases}\]

Find the points of discontinuity, if any, of the following functions:  \[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, & \text{ if }   x \neq 0 \\ 4 , & \text{ if }  x = 0\end{cases}\]

 


Determine if \[f\left( x \right) = \begin{cases}x^2 \sin\frac{1}{x} , & x \neq 0 \\ 0 , & x = 0\end{cases}\] is a continuous function?

 


If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then


The value of f (0), so that the function 

\[f\left( x \right) = \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}\]   becomes continuous for all x, given by

The points of discontinuity of the function\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]  


Show that the function 

\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at x = 1.


Discuss the continuity and differentiability of f (x) = e|x| .


Give an example of a function which is continuos but not differentiable at at a point.


If \[f\left( x \right) = \left| \log_e |x| \right|\] 


Let \[f\left( x \right) = \begin{cases}\frac{1}{\left| x \right|} & for \left| x \right| \geq 1 \\ a x^2 + b & for \left| x \right| < 1\end{cases}\] If f (x) is continuous and differentiable at any point, then

 

 

 


Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0

                     = k                    for x = 0

is continuous at x = 0. 


Discuss the continuity of f at x = 1 ,
Where f(x) = `(3 - sqrt(2x + 7))/(x - 1)` for x = ≠ 1
= `(-1)/3`   for x = 1


Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`


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


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


If Y = tan-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`


Discuss the continuity of the function f(x) = sin x . cos x.


Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if"  x ≠ 0),(0",",  "if"  x = 0):}` is discontinuous at x = 0.


f(x) = `{{:(3x - 8",",  "if"  x ≤ 5),(2"k"",",  "if"  x > 5):}` at x = 5


The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.


An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.


If f is continuous on its domain D, then |f| is also continuous on D.


Given functions `"f"("x") = ("x"^2 - 4)/("x" - 2) "and g"("x") = "x" + 2, "x" le "R"`. Then which of the following is correct?


Write the number of points where f(x) = |x + 2| + |x - 3| is not differentiable.


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