# Continuity and Differentiability - Class 12 - Mathematics

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## Examples - Continuity and Differentiability | NCERT | Mathematics | Class 12

Check the continuity of the function $f$ given by $f(x)=2 x+3$ at $x=1$.

To check the continuity of the function $f(x) = 2x + 3$ at $x = 1$, we need to verify three conditions:

The function $f$ is defined at $x=1$.

The limit of $f(x)$ as $x$ approaches $1$ exists.

The limit of $f(x)$ as $x$ approaches $1$ is equal to $f(1)$.

**Step 1:** Check if $f$ is defined at $x = 1$.

Since $f(x) = 2x + 3$ is a polynomial function, it is defined for all real numbers, including $x = 1$. Hence, $f(1) = 2(1) + 3 = 5$.

**Step 2:** Compute the limit of $f(x)$ as $x$ approaches $1$.

For polynomial functions, the limit at any point is simply the value of the function at that point. Thus, $$\lim_{x \to 1} f(x) = \lim_{x \to 1} (2x + 3) = 2(1) + 3 = 5.$$

**Step 3:** Check if the limit is equal to $f(1)$.

From our calculations, $\lim_{x \to 1} f(x) = 5$ and $f(1) = 5$, so they are equal.

**Conclusion:** Since $f(x) = 2x + 3$ meets all three conditions for continuity at $x = 1$, it is continuous at $x = 1$.

## Extra Questions - Continuity and Differentiability | NCERT | Mathematics | Class 12

Let $f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+19$. Then $f(x)=0$ has

A. 13 real roots

B. only one positive and only two negative real roots

C. not more than one real root

D. has two positive and one negative real root

The correct option is **C**: not more than one real root.

To determine the number of real roots of the function $f(x) = x^{13} + x^{11} + x^9 + x^7 + x^5 + x^3 + x + 19$, we first compute its derivative:

$$ f'(x) = 13x^{12} + 11x^{10} + 9x^8 + 7x^6 + 5x^4 + 3x^2 + 1 $$

Next, we observe that each term in the derivative $f'(x)$ is **strictly positive** for all $x \in \mathbb{R}$. Therefore, $f'(x) > 0$ for all real numbers $x$. This implies that the function $f(x)$ is **strictly increasing** across the entire set of real numbers.

Since $f(x)$ is a **strictly increasing function** and its highest degree term $x^{13}$ dictates its behavior at the extremes (as $x \to \infty$ and $x \to -\infty$, $f(x)$ behaves like $x^{13}$, approaching $\infty$ and $-\infty$ respectively), it follows from the Intermediate Value Theorem that there must be exactly **one real root**.

This is because a strictly increasing function can cross the x-axis at most once. Thus, we conclude that $f(x) = 0$ has exactly one real root.

Difference between integration and differentiation

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For a real number $y$, let $[y]$ denote the greatest integer less than or equal to $y$. Then the function $f(x)=\frac{\tan \pi([x-\pi])}{1+[x]^2}$.

A. Discontinuous at some $x$

B. Continuous at all $x$, but the derivative $f'(x)$ does not exist for some $x$

C. $f'(x)$ exists for all $x$, but the derivative $f''(x)$ does not exist for some $x$

D. $f'(x)$ exists for all $x$