3rd Funky Functions: C^1 Function.   

A function that is differentiable,  

but the derivative is not continuous.  

 

Let b>0 and consider the function:  

   f : [-a, +a] --> R  

   f(x) = x^b sin(1/x)   if x isn't 0  

   f(x) = 0                    if x = 0 .  

In Problem 5.1.13 we (basically,  

due to symmetry) showed that  

a) f is cont. at x=0  iff b>0  

b) f is diff. at x=0 iff b>1  

c) f '  is cont at 0 iff b > 2.  

 

In the Funky Functions assignment,  

the 3rd Funky Function function was  

the above function f with b=2.  

Let's see how this function f looks  

on the domain [-a, +a] with b=2.  

 

> restart;  
a:= .1;
b:= 2;
f := x -> x^b*sin(1/x);
plot(f(x),x=-a..a);
 

 

 

 

.1
2
proc (x) options operator, arrow; `*`(`^`(x, b), `*`(sin(`/`(1, `*`(x))))) end proc
Plot_2d
 

 

Ok,  now below you can mess around with  

the domain [-a, a] of f as well as the b>0  

and see what happens.    

 

> restart;  
a:= 0.01;
b:= 2;
f := x -> x^b*sin(1/x);
plot(f(x),x=-a..a);
 

 

 

 

0.1e-1
2
proc (x) options operator, arrow; `*`(`^`(x, b), `*`(sin(`/`(1, `*`(x))))) end proc
Plot_2d
 

>
 

If you want, you can also adjust   

the range (output) of the graph (plot) of f.  

> restart;  
a:= 0.01; d:= 0.00002;
b:= 2;
f := x -> x^b*sin(1/x);
plot(f(x),x=-a..a,y=-d..d);
 

 

 

 

 

0.1e-1
0.2e-4
2
proc (x) options operator, arrow; `*`(`^`(x, b), `*`(sin(`/`(1, `*`(x))))) end proc