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in Continuity and Differentiability by (27.3k points)
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Find the points of discontinuity, if any, of the following functions :

\(f(x) = \begin{cases}\frac{x^4+x^3+2x^2}{tan^{-1}x}&,\quad if\, x ≠0\\10&,\quad if\,x=0\end{cases} \)

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A real function f is said to be continuous at x = c, 

Where c is any point in the domain of f 

If :

\(\lim\limits_{h \to 0}f(c-h)\) = \(\lim\limits_{h \to 0}f(c+h)\) = f(c)

where h is a very small ‘+ve’ no.

i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if :

\(\lim\limits_{x \to c}f(x)\) = f(c)

Here we have,

\(f(x) = \begin{cases}\frac{x^4+x^3+2x^2}{tan^{-1}x}&,\quad if\, x ≠0\\10&,\quad if\,x=0\end{cases} \) …Equation 1

The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain 

(domain = set of numbers for which f is defined)

Let c is any random number such that c ≠ 0 

[thus c being a random number, it can include all numbers except 0]

∴ We can say that f(x) is continuous for all x ≠ 0 

As zero is a point at which function is changing its nature so we need to check the continuity here. 

f(0) = 10 [using eqn 1] 

And,

∴ f(x) is discontinuous at x = 0 

Hence, 

f is continuous for all x ≠ 0 but discontinuous at x = 0

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