Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let a function $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=\{\begin{array}{ll}\sin x-e^x& \text{if }x\le 0\\ a+[-x]& \text{if }0<x<1\\ 2x-b& \text{if }x\ge 1\end{array}$ Where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $\mathbb{R}$, then $(a+b)$ is equal to:

Imagine you are standing on a bridge that is being built from three different sections. For the bridge to be safe to walk across, each section must connect perfectly to the next without any gaps or sudden drops.

In mathematics, this is exactly what continuity represents. We are given a piecewise function $f(x)$ that changes its identity at $x=0$ and $x=1$. Our mission is to find the values of $a$ and $b$ that make this function a seamless, unbroken curve across the entire real number line.

At $x=0$, our function transitions from the blue curve defined by $f(x)=\sin x-e^x$ to the green segment defined by $f(x)=a+[-x]$. For continuity, the left-hand limit (LHL) must equal the right-hand limit (RHL).

Let's calculate the LHL as $x$ approaches $0$ from the left:

Now, for the RHL, we look at the green segment:

As $x$ approaches $0$ from the right, $x$ is a tiny positive number, so $-x$ is a tiny negative number (like $-0.001$). The greatest integer less than or equal to $-0.001$ is $-1$. Thus, the RHL is $a-1$.

Equating them, we get $-1=a-1$, which gives us $a=0$.

Now that we know $a=0$, our function in the middle interval $0<x<1$ is simply $f(x)=[-x]$. We move to the next junction at $x=1$.

The LHL as $x$ approaches $1$ from the left is:

Here, $x$ is slightly less than $1$ (like $0.999$), so $-x$ is slightly greater than $-1$ (like $-0.999$). The greatest integer less than or equal to $-0.999$ is $-1$. So, the LHL is $-1$.

For the RHL at $x=1$, we use the third piece of our function: $f(x)=2x-b$. Evaluating this at $x=1$, we get:

To ensure continuity, we set the LHL equal to the RHL:

Solving for $b$, we find $b=3$.

We have successfully navigated the junctions! We found $a=0$ and $b=3$.

The question asks for the sum $(a+b)$, which is:

By ensuring the limits matched at every transition, we have effectively welded these three mathematical segments into one continuous, elegant curve. Remember, in calculus, continuity is all about the connection—if you can bridge the gap between the left and the right, you have mastered the concept. The final answer is $3$.