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Continuity and Discontinuity - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Continuity is considered one of the most asked concept.

  • 257 Questions around this concept.

Solve by difficulty

If the function.

g(x)=   \left \{ \right. 

is differentiable, then the value of k + m is :

Consider the function.
$f(x)= \begin{cases}\frac{a\left(7 x-12-x^2\right)}{b\left|x^2-7 x+12\right|} & , x<3 \\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\ b & , x=3\end{cases}$

where $[\mathrm{x}]$ denotes the greatest integer less than or equal to $\mathrm{x}$. If $\mathrm{S}$ denotes the set of all ordered pairs $(\mathrm{a}, \mathrm{b})$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is:

Concepts Covered - 1

Continuity

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)  \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.\;\;\text{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.

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Continuity

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