title: Smooth connection between piecewise parts
body: <p>Example piecewise wise function:</p>
  
  <pre><code>f[x_]:=Piecewise[{{x^2, 0&lt;x&lt;1-epsilon},{x,1&lt;x&lt;2-epsilon},{2,x&gt;2}}]
  </code></pre>
  
  <p>Is there a way to connect these parts in interval epsilon, so I get a smooth function?<br><br>
  EDIT: <br>By smooth, I don't mean it needs to be derivable in point of connection, just that in some numerical work it looks like a "natural" connection.<br><br>
  EDIT2:<br>
  Two black circles represent the points where lies the problem. I'd like it to look like a derivable function (although it doesn't need to be in rigor mathematical sense, but I don't want these two spikes). Red circle represents the part where it looks good.<br>
  <img src="https://i.stack.imgur.com/57DiD.png" alt="picture">
  <br>What I could do is do this by nonlinear fitting the [x-epsilon, x+epsilon], but I was hoping that there was an easier way with piecewise function.</p>