Animated Solution for Mathematics - Limits, Continuity and Differentiability: The function $f(x)=1+|\sin x|$ is

The function is defined as $f(x)=1+|\sin x|$. To understand its behavior, we first visualize the parent function $g(x)=\sin x$.

The absolute value operator acts as a mirror, reflecting all negative portions of the sine wave above the $x$-axis. This transformation creates a series of sharp, rhythmic peaks at every point where the sine function previously crossed the axis.

Continuity is the property of a function being unbroken. When we trace the sine wave, the pen never leaves the paper.

Applying the absolute value simply folds the negative segments upward. Because the sine function is continuous everywhere, this reflection process does not introduce any jumps, holes, or asymptotes.

Therefore, the function $f(x)=1+|\sin x|$ is continuous for all real numbers.

Differentiability requires the existence of a unique tangent line at every point, which corresponds to the "smoothness" of the curve. At the peaks of our reflected sine wave, we encounter a 'V' shape, known as a cusp.

To prove this rigorously, we examine the inner function $g(x)=\sin x$. The modulus function $|g(x)|$ fails to be differentiable at points where $g(x)=0$, provided that the derivative $g'(x) eq 0$.

For our function, $g(x)=\sin x=0$ at $x=n\pi$, where $n$ is any integer. The derivative is:

At these points, the value of the derivative is:

Since the derivative is non-zero at these roots, the slope changes abruptly from positive to negative. This confirms the existence of a sharp corner at every $x=n\pi$.

The vertical translation by $1$ unit does not alter the slope or the sharp nature of these corners; it merely shifts the entire graph upward.

Because the sine function has roots at every integer multiple of $\pi$, the function $f(x)=1+|\sin x|$ is not differentiable at $x=n\pi$ for all $n\in \mathbb{Z}$.

In all other regions, the function remains smooth and differentiable. You have successfully identified that the function is continuous everywhere but possesses an infinite set of points where it fails to be differentiable.