Is the Canon EF-S 55-250mm f/4-5.6 IS II Lens sharp? This page visually answers that question. Shown above are 100% crops from a lab-captured 100mm f/11.0 test image. Observe the contrast and resolution of these results.
Is the Canon EF-S 55-250mm f/4-5.6 IS II Lens set to 100mm f/11.0 sharper than the Canon EF 75-300mm f/4-5.6 III Lens set to 75mm f/5.6? The image quality comparison above also answers that question. Switch between the results to see the sharpness differences between these lenses.