Consider the piecewise linear function defined by $$y_0(x)=\begin{cases}0&x\in[-1,0]\\x&x\in(0,1]\end{cases}$$ I want to find a sequence of function in the domain $$D=\{y\in C^1[-1,1]\big|y(-1)=0,y(1)=1\}$$ that uniformlly converges to $y_0$.
Generally, when given a piecewise linear function $y_0$ defined on $[a,b]$, whether there exists a sequence of functions in $D=\{y\in C^1[-1,1]\big|y(a)=y_0(a),y(b)=y_0(b)\}$ converges uniformly to it?