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 $\lim x \rightarrow a f(x)$ does not exist. We must add another condition for continuity at $x=a$, which is
(ii) $\lim _{x \rightarrow a} f(x)$ 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) $\lim _{x \rightarrow a} f(x)=f(a)$

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. $\lim _{x \rightarrow a} f(x)$ exists
3. $\lim _{x \rightarrow a} f(x)=f(a)$

Or

$
\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=f(a)
$

i.e. L.H.L. $=$ R.H.L. $=$ value of the function at $x=a$

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