Animated Solution for Mathematics - Limits, Continuity and Differentiability: The function $f(x)=\max \{(1-x),(1+x),2\},x\in (-\infty ,\infty )$ is

Unveiling the Max Function

Imagine you are walking along a mountain range. At any point in your journey, you are standing on the highest peak available to you.

This is exactly what the function $f(x)=\max \{(1-x),(1+x),2\}$ represents. It is a decision-maker, a selector that constantly chooses the largest value among three competing paths.

To master this, we must stop thinking of it as a scary algebraic expression and start seeing it as a landscape.

Let us introduce our three contenders. First, we have $y_1=1-x$, a line descending with a slope of $-1$. It is a steady decline, passing through $(-1,2)$ and $(0,1)$.

Second, we have $y_2=1+x$, a line ascending with a slope of $1$, passing through $(0,1)$ and $(1,2)$. Finally, we have the steady, unwavering $y_3=2$, a horizontal line that refuses to change regardless of $x$.

To find $f(x)$, we are looking for the 'upper envelope.' Imagine a spotlight shining from above; the graph of $f(x)$ is the shadow cast by the highest line at any given $x$.

As we move from $x=-\infty$ toward the right, we see that $1-x$ starts very high. But as $x$ increases, $1-x$ drops. Meanwhile, $2$ stays constant.

At $x=-1$, the line $1-x$ hits the value $2$. Before $x=-1$, $1-x$ is greater than $2$. After $x=-1$, the constant $2$ becomes the dominant force.

Similarly, as we approach $x=1$, the constant $2$ is still the highest, but once we cross $x=1$, the line $1+x$ begins to climb higher than $2$. Thus, our function reveals its true piecewise nature:

Continuity is the soul of a graph—it asks, 'Is there a break in the path?' Let us check the transition points.

At $x=-1$, the left-hand limit is $1-(-1)=2$, and the right-hand limit is $2$. They meet perfectly!

At $x=1$, the left-hand limit is $2$, and the right-hand limit is $1+1=2$. Again, they meet. Because the pieces connect without any jumps, the function is continuous everywhere. You can draw this graph without ever lifting your pen from the paper.

Now, we reach the most exciting part: differentiability. Differentiability is about smoothness. A function is differentiable only if it has a unique, well-defined slope at every point.

At $x=-1$, the slope to the left is:

But the slope to the right is:

Because $-1 eq 0$, the graph has a sharp 'kink' at $x=-1$. The same happens at $x=1$, where the slope changes from $0$ to $1$.

These sharp corners are the 'traps' of calculus. They are continuous, yes, but they are not smooth. Therefore, while the function is continuous at all points, it fails the test of differentiability at $x=-1$ and $x=1$.

By visualizing the function as an upper envelope, we transformed a daunting algebraic problem into a clear geometric story.

We learned that continuity is about connection, while differentiability is about the grace of a smooth curve. You have successfully navigated the landscape of this function, proving that with the right perspective, even the most complex piecewise definitions become intuitive and elegant.