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Jamia Millia Islamia MCA Continuity PYQ


Jamia Millia Islamia MCA PYQ
If $f(x)=x^{2}\sin\frac{1}{x}$ for $x\neq0$, then the value of the function $f$ at $x=0$, so that $f$ is continuous at $x=0$, is





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2021 PYQ

Solution


Jamia Millia Islamia MCA PYQ
Which statement is false?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2022 PYQ

Solution

Continuity of $f(x)$ at $x=a$ does **not imply** that its inverse $f^{-1}(x)$ is continuous there (the inverse may not even exist). $\boxed{(B)}$

Jamia Millia Islamia MCA PYQ
The function $f$ is defined in $[-5,5]$ as $ f(x)= \begin{cases} x, & \text{if }x\text{ is rational}\\ -x, & \text{if }x\text{ is irrational} \end{cases} $





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2022 PYQ

Solution

For rationals $f(x)=x$, for irrationals $f(x)=-x$. At $x=0$, both give $0$, so it’s continuous there. At any $x\neq0$, limits from rationals and irrationals differ.

Jamia Millia Islamia MCA PYQ
A function is continuous at x = a if:





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2025 PYQ

Solution

Continuity condition
$\lim_{x \to a} f(x) = f(a)$

Conditions:
$f(a)$ exists
$\lim_{x \to a} f(x)$ exists


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