Mathematics I (Calculus) - Old Questions

6. Define continuity on an interval. Show that the function  is continuous on the interval [ -1,1] . 

A function f is continuous from the right at a number a if  and f is continuous from the left at a if  .

A function f is continuous on an interval if it is continuous at every number in the interval. If f is defined only one side of an end point of the interval, we understand continuous at the end point to mean continuous from the right or continuous from the left.

Given,

Let  then

Which shows that f(x) is continuous at .

For the end points i.e. x=-1

Which shows that f(x) is continuous at the left end point x = -1

Similarly for the end point x=1

Which shows that f(x) is continuous at the right end point x = 1.

Hence f(x) is continuous at [-1,1]