Animated Solution for Mathematics - Limits, Continuity and Differentiability: The function $f(x)=\{\begin{array}{ll}\frac{\pi }{4}+{\tan }^{-1}x,& |x|\le 1\\ \frac{1}{2}(|x|-1),& |x|>1\end{array}$ is:

To understand the behavior of the function, we first express it without the absolute value notation. The function is defined as:

By expanding the modulus across the critical points $x=-1$ and $x=1$, we obtain the explicit piecewise definition:

Continuity requires that the limit from the left equals the limit from the right at the transition points. We first examine the seam at $x=-1$:

Since $LHL=RHL=f(-1)=0$, the function is continuous at $x=-1$.

Next, we examine the seam at $x=1$:

Because the $LHL eq RHL$, the function has a jump discontinuity at $x=1$. Therefore, the function is discontinuous at $x=1$.

Differentiability requires the function to be continuous and possess equal left-hand and right-hand derivatives. Since the function is discontinuous at $x=1$, it is automatically non-differentiable at $x=1$.

We now evaluate the derivatives in the open intervals:

At $x=-1$, we compare the one-sided derivatives:

Since $LHD eq RHD$, the function has a sharp corner at $x=-1$. Thus, the function is non-differentiable at $x=-1$.

By analyzing the seams and the slopes, we conclude that the function is continuous on the domain $\mathbb{R}-\{1\}$. Furthermore, the function is differentiable on the domain $\mathbb{R}-\{-1,1\}$.

Always remember that while continuity is a prerequisite for differentiability, it does not guarantee it. Sharp corners and jump discontinuities are the primary obstacles to a function being "smooth."