Continuity and Discontinuity - Practice Questions & MCQ

Continuity

Many functions have the property that their graphs can be traced with a pen/pencil without lifting the pen/pencil from the page. Such functions are called continuous. Other functions have points at which a break in the graph occurs, but satisfy this property over some intervals contained in their domains. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs. So continuity can be defined in two ways: Continuity at a point and Continuity over an interval.

Continuity at a point

Let us see different types of conditions to see continuity at point x = a

We see that the graph of f(x) has a hole at x=a, which means that f(a) is undefined. At the very least, for f(x) to be continuous at x=a, we need the following conditions:

(i) f(a) is defined

Next, for the graph given below, although f(a) is defined, the function has a gap at x=a. In this graph, the gap exists because lim x → a f(x) does not exist. We must add another condition for continuity at x=a, which is

(ii)   exists

The above two conditions by themselves do not guarantee continuity at a point. The function in the figure given below satisfies both of our first two conditions, but is still not continuous at a. We must add a third condition to our list: 

(iii) 

So, a function f(x) is continuous at a point x = a if and only if the following three conditions are satisfied:

1. f(a) is defined 

2. exists

3. 

Or

A function is discontinuous at a point a if it fails to be continuous at a.