Graphing a Function with the Given Properties

Question

I’m having trouble drawing a function that has these properties. In particular, the continuity and differentiable parts. The picture provided takes cover of (a) and (d). Thanks for the help.

Answer 1

Recall that a jump discontinuity at $x = a$ exists whenever $\lim_{x \to a^-} f(x) = L^-$ and $\lim_{x \to a^+} f(x) = L^+$ both exist and yet $L^- \neq L^+.$ Consequently, one can visualize these (quite literally) as a jump in the function from one point on the left-hand side to a distinct point on the right-hand side. Be careful when writing the jump discontinuities to choose your values so that you do not violate the (odd) symmetry of $f(x).$

Recall that $f(x)$ is differentiable at $x = a$ whenever the limit $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)} h = \lim_{h \to 0} Q_a(h)$$ exists. Consequently, if you wish for $f(x)$ to not be differentiable at $x = a,$ you can force a jump discontinuity in $Q_a(h)$ as $h$ tends to $0.$ Particularly, you may take $\lim_{h \to 0^-} Q_a(h) \neq \lim_{h \to 0^+} Q_a(h).$ Graphically, this amounts to forcing a cusp or a corner in your function at $x = a$ since the function $Q_a(h)$ gives the slope of the secant line of $f(x)$ over the interval $[a, a + h].$ Once again, be careful not to mess up the symmetry.

Answer 2

For example,

\begin{align} f(x)&= \begin{cases} -\sin(\tfrac\pi2\,x),\quad & x\in [0,1) ,\\ 3-x,\quad &x\in [1,2) ,\\ (-2x+4)\exp(-x+3)+1,\quad &x\in [2,\infty) ,\\ -f(-x),\quad & x<0 . \end{cases} \end{align}